Certainly something just across the river near MRT stations 10k NTD probably will not get you a building you want to live in. I moved from Taipei to New Taipei City (but I was looking for 2-3 bedroom) and most places I saw did not include water, gas or electricity. A few places included TV/Internet. Usually they add on the management fee on top of the rent but you might be able to negotiate that. (i’m lucky that I had people that helped me talk to the landlord as most don’t speak english)
Buildings that are 30yrs+ in age normally don’t have any management fee and rent is much cheaper due to no elevator / no security / no trash service.
I would use Google Chrome or some browser with “translation” function to search rent.591.com.tw to get a better idea. You can use the “map search” function to go to your preferred neighborhood on the map and then adjust the price to NT5000-10000 to see what is available. Lots of pictures and although the translation to English is a mess you can get an idea of what is included. I would guess that for something relatively close to a MRT, 10k places will be very small, 30yrs+, probably a tiny fridge, and a hot plate.
(Back in 2003 rent for a large studio, with all appliances, in a security guard building with trash service in Tu-Cheng district was already about NT 8000-9000. This was way BEFORE the MRT opened up.)