I’m not getting it. To me, the first vowel in Kiri/Kyra/Keira have the same ee sound as bleary. Maybe the OP is using “Kiri” to represent a different vowel sound (there is a fifth sound people use besides Mary/merry/marry/meerkat to represent “Carrie/Kerry” but I can’t think of how to show it).
Actual conversation I had with a West Texas girl when I was in collge.
Her: My name is Va.
Me: Va?
Her: No! Va!
Me: Va?
Her: No! Va!!
Me: Va?
Her: No! Va!!!
Me: I have no idea here.
Her: Va! Va! As in Va-Oh-La (Viola)
Me: Oh! Vi!
Her: Yes Yes, Va!
But it is not the sound of the actual vowel itself, it is the attack (start) of the vowel sound. Most Americans, I think, add the “y” attack to certain vowel sounds. The similar word “carry” tends to be pronounced like “kyarry”, compare that to “car”, which has a clean, hard attack. Guttural consonants – c, k, g – draw the tongue into a position that makes the “y” sound a natural transition into the palate-formed vowels.
My Australian friend once wrote this down and asked me to say it: “Rise Up Lights”
She then said “You have just said ‘razor blades’ in an Australian accent.”
That fifth sound can be something close to short “i” in General American (Newscaster Speak) “mitt”. That’s my go-to vowel for the first syllable in “Kiri” if no one indicates otherwise. For what it’s worth, that’s how you’ll usually hear Kiri Te Kanawa’s first name most commonly spoken.
Now write down and say: PSDS
You have just said “pierced ears” in a Boston accent.
Yes - thanks. But I say Kiri Te Kanawa as “bleary” – that’s what I was channeling upthread.
I would say I attack Carrie/carry and car the same way - k and then the vowel sound. But at this point I could be wrong!!
Now say “chester draws”, and you have “chest of drawers” in Boston.
I go through this all the time with my own name. The form of my first name that I use has 3 letters: 2 consonants separated by a vowel. Simple. But when I just say my name, people tend to think I’m saying something else. So I have to spell it. Like:
*What’s your name?
Dan.
Den?
No, Dan.
Din?
No, Dan, D-A-N.
Oh, Dan!
*
And don’t even ask me about my last name:
*What’s your last name?
Washington.
Well, Mr. Morningschteiner . . .
Do you spell “wash” with an “M”? Do you pronounce “ton” “schteiner”?
I’m sorry, it’s just that “Washington” is such an unusual name.
*
Also know as “Strine”.
We don’t say “drawer” and “draw” exactly the same. It’s subtle, but different. But we obviously don’t pronounce drawer the way you’re supposed to. Borrower either.
Dawn and Don, taught and tot, caught and cot, those we actually do pronounce exactly the same. Some people claim they saw Dawn and Don differently, but it always sounds like “That’s not true, I say don and don!” yeah.
Pairs like Erin and Aaron are mixed, some people pronounce both like other people pronounce Erin, and others slightly different. I was made aware of this when I realized I could sometimes tell without context if people mean my college friend Erin or the boy Aaron she was dating at the time.
I can hear the difference, but it is so not-ingrained that I cannot produce it most of the time. “Dawn” is what sounds normal to (the mid-country folk), to say “don” so they know you do not mean dawn, you say a vowel that sounds like a mix of “ah” and a short a (as in “bad”), like you are pushing the vowel toward your palette. To me, it seems kind of backwards, an “o” vowel sound should not be flattened like that, “o” is round for a reason.
To me, “caulk” sounds exactly like a word for rooster. At least, the way most of us 'round here say it.
I once had an American friend ask a Scottish friend where he was from. He’s from Stirling, but in his accent it came out something like “StaeRRRling”. I could hear the difference and he could hear the difference, but she (my American friend) couldn’t. Which led to this conversation:
He: I’m from Stirling [StaeRRRling].
She: Starling?
He: Nae, StaeRRRling.
She: Starling, with an “A”?
He: Nae, with an “aaaeee”!
This is fascinating to me! My Aussie husband does NOT recognize other accents (dialects) at all. It’s bizarre to me. I’m always translating for him when we watch “Outlander” or anything other than Aussie or British. When we were first dating, I had a conversation with a shopkeeper with a heavy Cuban accent and he was impressed and said “I didn’t know you knew Spanish!” Uh, I don’t, the shopkeeper was speaking English!
When we lived in Singapore, I used to watch a terrible local sitcom in which they spoke typical English/Singlish to get used to the sounds and he came in one day and wondered why I’d be following a show that’s in Mandarin.
Which also reminds me of my NYC friend. We were about 23 and debating the pronunciation of “gnocchi” and asked the Italian waiter how he said it. He replied and we turned to each other and said, at the exact same time, “See?!” 
This may be the thread for a spelling peculiarity that has always interested me.
In Oz/UK/(other places possibly?) the informal term for mother is spelled “Mum”
In the US, it’s spelled “Mom”
Those spelling differences seem to mask an underlying uniformity - because of the vagaries of vowel pronunciation, an American saying “Mom” and an Aussie saying “Mum” would actually sound … pretty much the same, really.
I can’t think of any other words this would apply to.
I’m from Philadelphia, so to me Mary, marry, and merry have always had three very distinct pronunciations. (Over the summer I dated a girl from Chicago named Mary, who teased me about the way I pronounced “marry.”)
I have a question for those who pronounce those three words the same way. For each word, I can think of a number of rhyming words and/or words which use the same vowel sound:
Mary:
airy
hairy
dairy
fairy
vary
pair, pear
rare
stare
tare
bear
care
wares
marry:
Larry
tarry
carry
Harry
Barry
parry
Gary
merry:
Jerry
Terry
berry
very
ferry
Do people who pronounce Mary, merry, and marry the same, pronounce the first vowel sound of all of the above words the same? If not, for those who say they cannot hear or understand the distinction, does thinking about the difference between all of those words help you understand?
Yes, in my Texas dialect, all of those words have the same exact vowel sound.
Yes. Every one of the words you list rhyme perfectly to me. So no, thinking about the difference between all those words doesn’t help me understand. Because to me, there simply is no difference.
That’s dialects for you. Frustrating, isn’t it? 
This. Born in California, lived all over, Midwest for the last 20-ish years, and now Miami.
All 60-ish of Arcite’s words are pronounced exactly the same by me. And almost always sound to me as though other people pronounce them exactly the same too.
For some people some times I can hear them slightly making a MAAAAARRRRRRy sort of noise that is subtly different. It sounds more like an affectation than a genuine difference though. And I can’t tell you now which of the three word sets has this subtly different sound.
Yes, I desperately want to help you see the light.
I think I can sort of understand not being able to make those three different sounds yourself, since we know you really have to learn how to make a particular sound in childhood to make it well. What I don’t get is how someone cannot even hear the difference, even when demonstrated by someone pronouncing the three different words in a deliberately exaggerated manner, with exaggerated mouth movements as well (as someone upthread said they did,) so as to highlight the differences. They’re three different sounds. How can you hear them all the same way?
(I guess I’m someone with a knack for languages. When they gave us that Pimsleur test in the 8th grade, I got the highest possible score. I’ve had speakers of different dialects or languages explain or demonstrate various subtleties of sound in their language to me, and while I often can’t reproduce the sounds accurately, I’m always able to hear them.)