We used to call that one “Asshole/President”. First one out was President, the guy at the end was the asshole. Don’t ask me the logic behind that one.
“Bullshit!” is another favorite.
Err, with modified titles, of course.
We used to have a game called Asshole in high school, but it was different from Shithead (which, if the OPs daughter plays Neopets, she might already know as Go Go Go.) And I see good ole Pagat has the rules. We only had Pres, Ass, Vice-pres and vice-ass designations though, with the Pres and Ass swapping two cards, the Vices one, and everyone else called ‘neutral’. We didn’t always switch seats either.
I know another one with a name I won’t mention here, just give it any four-letter word for a title (Trust me, swears are very appropriate for this with an older audience–you’ll be saying them anyways!). Goes very fast once you get going, but needs at least four people to have a good game.
You take out four cards for each person playing–at least the Aces, Kings, Queens and Jacks, and keep going lower for more players. These are the cardsyou play with. Each person is dealt four cards. The goal is to collect all of a rank of cards. You pick one card fromyour hand, and place it to your right, and then pick up the card that has been laid to your left. You can’t place down another card until the first one has been picked up, and you can’t pick up another card until you’ve played one down.
When someone collects their four cards, they place their hand with the cards in the middle. Then, everyone has to place a hand on top of theirs, only they can’t be holding any cards in that hand… The person whose hand is on top, or anyone who put has cards in their hand besides the first person gets the first letter of the title. Then you play another round. If the same person loses twice, they get the next letter in the word. (So, the first time you lose you have an ‘f’, if you lose again you have ‘fu’ and so on, or whatever you decide to call it.) Each loser earns a letter, and when they have the four letters of the title, then they lose.
“Bullshit!” is really “I Doubt It” with the order reversed.
If you don’t want to limit her to the standard deck of cards, consider buying UNO (a lot of fun, and available in lots of kid-friendly varieties now, incl. SpongeBob SquarePants, Harry Potter, The Simpsons, etc.), Rat-a-tat-Cat (a good memory and strategy game), and Loot (a pirate game in which you try to capture ships and gather treasure).
Our 9- and 6-year old sons like all of these, and they’re fun for grownups, too.
When I was that age, I played canasta and push rummy with my grandmothers. Was lots of fun. 
Cheat is good. If it’s just you and her though, don’t deal all the cards or it’ll be pretty boring.
Cheats is a simple game that any kid can play.
I think that ** Knockout Whist ** was the first card game I ever learnt.
You could always introduce her to the joys of gambling (for matches) with 3 card brag.
We used to play rummy with my mom when we were grade school age. Called it 500 rummy since that’s what we played to. You have to agree on the value of the aces, though. At our house they were always 20 points, but some people play them as 15, and some say it depends on how they’re played, high or low.
You’d have to get a deck for it, but my great-nieces love Dutch Blitz. That might not be a great idea with just two of you, though. It’s really better with more players.
I’ll second Rummy and Gin Rummy, although my family never did play for money (the games were vicious enough as it was). Oh Hell was also a family favorite, but, since Mum doesn’t swear, we called it “Up and Down”. A commercial version of this is Wizard and I highly recommend it since it removes one of the flaws of Oh Hell.
You might also want to try a game called Spoons which Mum taught her grandkids when they were about 5. You need at least three players for this. What you do is take one rank of cards per person playing, i.e. Jacks, Queens, and Kings if 3 people are playing, i.e. Jacks, Queens, Kings, and Aces if 4 people are playing, etc. and one fewer spoon than there are number of players. I recommend old spoons and old cards. Shuffle the cards and deal them out and place the spoons in the middle of the players. Each person selects a card from her hand and, on a preset signal, passes it to the player on her right. As soon as someone gets four of a kind, she puts her cards down so everyone can see them and grabs for a spoon. The person who doesn’t get a spoon gets a letter. The game is over when someone has enough letters for Donkey or whatever word you’ve settled on. This was one of my favorite games as a kid and I still enjoy playing it with my brother’s kids. We have had tug-of-wars over spoons (hint: if that’s the case, the person with the bowl end has a large advantage); when we were kids, my brothers and I also tore a few cards and bent a few spoons, which is why this game is best played on the floor.
Enjoy!
CJ
Isnt that The Great Dalmuti / Corporate Shuffle ?
Again, if playing with non-standard pack of cards equipment is allowed, Uno Attack is a fantastic version of the old favorite. It’s really Uno combined with Russian Roulette. Fun for kids and adults.