That is, if I were to eat the same thing every day for a year, I’d get sick of it and lose my appetite when lunch time came around, but I’ve slept every single day of my life and I’m not sick of that?
Certainly both eating and sleeping are necessary for my continued existence…
Perhaps a grander concept for that question is: why do some things become boring and repetitive while others don’t?
Because you’re not sick of eating per se, but rather the taste of the food.
Not being a physiologist nor a physcohologist this just my view.
Sleeping is a necessary restorative process or (lack of ) physical activity. 
Eating is a necessary activity to maintain healt/activity/growth/etc. Something of a stimulative activity.
One song, meal, physical activity, etc. gets boring because it is repetitive stimulus. 
There is probably an evolutionary advantage to eating a variety of foods. The desire to seek out new foods and the overall hunger drive itself were instrumental in ensuring that people survived long enough to reproduce. The stronger the hunger drive, and the greater the desire to eat different foods would stave off starvation and disease.
Also, getting rest, and preserving one’s energy would also be an evolutionary advantage.
because eating one thing is evolutionarily detrimental, thus we have evolved to get bored of a constant diet. Sleeping in one style is not so we havent.
Tell that to the koala.
I forget the exact number, but my nutritionist-grad-student fiancée was telling me just the other day that most people only eat about 20ish different foods on a regular basis, and the only time you will expand the variety of foods you eat is if your body isn’t getting enough of some vital nutrient. So if you’ve been eating chicken nuggets and glazed donuts and beer for the last month, and someone says, “hey, want to go to So-and-so restaurant?” you will almost certainly think of something they have to offer that contains lots of different kinds of food – like a mixed green caesar salad with sunflower seeds and ham and sliced egg and so forth. Your body seeks out variety to get the nutrients it needs.
I’m going to go have a burrito now.
This is an apples-and-oranges comparison. While it’s true that one would get tired of eating the same thing every day, it’s not true than one gets tired of eating at all. It’s hard to imagine what the sleep equivalent would be of eating a varied diet. What would you do - sleep in different positions? In different beds?
Deciding what to eat is a conscious process, and awareness of what one is eating also involves consciousness. Although one can will oneself to stay awake, nothing in the act of sleeping itself involves consciousness decision-making or awareness. Sleep is a physiological process, like digestion. Why don’t people get tired of digesting their food every day?
But they sure look bored as hell . . .
As Philster suggests, for an omnivore like a human getting “tired” of one food is most likely an adaptation to ensure that we get sufficient variety to acquire all various nutrients and vitamins we need.
It may also help prevent taking in too much of toxic substances that might be present in small amounts in certain foods. Howler monkeys, for example, will vary their diet so they don’t have to deak with too much of certain toxic chemicals found in leaves of particular species.
More specialized animals, whose are able to acquire everything they need from a more limited diet, most likely lack this response, and are perfectly happy eating the same thing every day.
Maybe you’d get tired of sleeping if you had the same dream every night. 
Depends on the dream!
But they’re an endangered species, aren’t they?
The other thing to remember is that most of us don’t really get that hungry. Not starving hungry. We eat when our stomach is empty or because it’s time to eat, and because we’re so well fed, the same old same old doesn’t have the same old appeal.
Starve a person for a couple of days, and they’d be delighted to get their hands on the same old once more.
I’ve been thinking about this, and I don’t think this is quite right. Clearly, we could have evolved to have the ability to acquire all our nutritional needs from a single source, the way the koala did. But we didn’t. Why? Because being dependent on a single plant or animal species for nutritional requirements is risky. If a disease swept through the eucalyptus population, wiping it out, the koala is screwed unless some of them can adapt to a new diet. Additionally, having a more extensive diet ensures that suitable nutrition can be found year-round, in a wide variety of climates - another advantage, evolutionarily speaking. the other things you mention are certainly benefits of a highly variable diet, but not the root evolutionary cause, IMO.
What Philster and Q.E.D. said.
Google for “specialists versus opportunists” to find more thoughts on these two different evolutionary strategies.
Specialists make the most of a foodsource that is plentiful and often unattractive to others. They are vulnerable when that foodsource disappears, like the koala when his eucaliptustrees are struck down, or the panda when his bambooforests disappear. Generally, these animals don’t have much curiosity or “drive”. Once they had their fill, they are perfectly happy sitting on their backsides, doing nothing. Specialists in the animal kingdom rarely get bored, and so it isn’t very cruel to keep them in cages.
Opportunists, OTOH, can eat many different variety’s of food, and have an innate drive to explore their surroudings for other things they might be able to eat. They survice in rapidly changing enviroments, because they will seek out (and find) other foodsources. They are curious, driven, inventors. The raccoon is the prime example, and so are primates. Opportunists get truly unhappy and bored when confined to a cage, even if they have enough to eat. Boredom triggers more exploration, and when you can’t explore, you get frustrated. That’s why man craves variety in foods.
Many omnivores are opportunists, but you can be an vegetarian or carnivore and still be an opportunist, or specialist, for that matter.
Here is a nice article on specialists and opportunists within the shark family.