I have a red eared slider turtle she's about 5 months old. She is in a 10 gallon aquarium for now(she's still small). I use a zoo med repti tuff splash proff halogen lamp 75 watt. Also a blue night light. I use a tetra 10 to 20 gallon reptile safe filter. When I first got her, she didn't really eat much. When she started eating, I think she would've ate anything. Now, it's been three weeks since she has eaten. She seems to be in a deep sleep. I am really worried about her. Hoping you can tell me what's wrong.
While I don't think the tank size is the problem, a 10 gallon tank is too small for even a hatchling, so as soon as possible I'd upgrade to a larger tank. Best would be at least 40-50 gallons, so that it will work for another few years. An adult RES would generally need a 100+ gallon tank. Also keep in mind that the filter should be able to handle *at least* twice the tank capacity.
A turtle cannot properly hibernate at house temperatures, so you are right to be concerned. You didn't say what your basking/water temperatures are, so check those first. Basking should be about 90 degrees (on the basking surface right under the lamp), and water about 76 degrees. If the turtle is too cool, she won't eat, but won't be cool enough for her metabolism to enter a true hibernation. This presents a real danger of starvation over time. The second issue is UVB. The basking bulb you're using is not a UVB bulb, and turtles need UVB in order to metabolize calcium. I would get a ZooMed Powersun 100w bulb--this will provide both heat and UVB. Keep in mind that it should be changed yearly. You don't need to provide night heat unless your house temperature drops below 50 degrees. You may have to play around with the height of the Powersun to get temperatures right since the tank is so small; if you have trouble, you can always put her in a larger plastic bin until you can get a bigger tank.
You also didn't say what her diet has been, but make sure to provide a variety of pellets, animal protein, and greens. Providing a cuttlebone for extra calcium as she needs it is also a good idea.
I'm attaching some links for further reading. If she still doesn't perk up after getting the Powersun and adjusting any temperature issues, please post back and let me know.