How do I get rid of these tiny moths in my room?

It sounds like pantry moths. If it is, they are fairly easy to get rid of if you can find the source. Look very carefully at the hampster’s food, as that could be the source and/or the attraction. You can just squash them or purchase a sticky trap. Look for signs of cocoons, and clean them away. Keep all food and seeds in airtight containers, and you should be fine.

Flood the room with insect spray, seal off the room and come back in 20 minutes. Then open all windows and scour the room with a vacuum cleaner to remove the corpses.
might want to remove the hamster first, too

Try to put a much brighter big light outside your window?

My kids have guinea pigs, and the bedding sometimes comes with bugs in it. Teeny-tiny flying bugs (I have no idea if they are moths). I would also keep the bedding in an airtight container.

It sounds like the bugs are in the hamster food or bedding.

Here’s are two suggestions: Buy cedar chips to put in the bedding. I don’t know if they’ll kill them, but they will prevent the moths from hanging around to lay more eggs. Or store bedding and food in the deep freeze. The colder it is, the better chance you will have of killing the eggs.

I have used a “fly tape” from a veterinary supply store. Hung it up with a bright light shining on it for a couple of nights.
Use seal containers for all food and bedding. Flour moths will show up in bird seed too.

Moth traps

DO NOT flood your room with insecticide, please. I read the comment above, and honestly that will not help the source. I have heard rosemary is a repellant for moths, but as others have said it is probably a flour/pantry/indian meal moth that is living in the hamster food or bedding. They usually like seeds so go through and look in the food if there is an infestation just throw it out. This is not a sanitation issue, humans and animals have lived together for hundreds of years. Many people will run to a disinfectant to clean everything after something like this, but it will only make them feel better. The disinfectant will not help your problem. Just look for larva and throw out the contaminated supplies so the population does not grow. You can also wipe down any surface with soapy water as that will kill any stray eggs. The soap will decrease the surface tension of the water and drown the egg/stragglers. Disinfectants will not do this. And once you have cleaned then put all food and or bedding into some type of sealed plastic container to avoid another infestation.

Moth eggs might also be in the tiny crevasses of the cage. You will need to thoroughly wash the cage with the hottest tap water you can get to kill the eggs, or perhaps you can clean the cage, and stick it in a deep freeze for a few days or a week. If you don’t have an alternative cage, the cage, the deep freeze would not work.