When I was going to do a program in Taipei for the summer, the people in charge of planning could only suggest AirBnB for accommodation. I was actually surprised at what I found… There are quite few apartments for rent within 5 min walk of an MRT w/private kitchen and bath for under NT25,000/month. (similar spaces will go for 30,000+ on 591找房). That’s above your budget, but I have high expectations for my living situation (and prefer to cook my own food in a kitchen and not on a hot plate on the floor of the moldy bathroom)
That being said, I have have only nightmare stories about living in shared living situations in Taiwan. I’ve never come across landlords and fellow tenants with so little regard for the law or even human decency.
At 13000/month, you can probably find a 套房 (that’s a room with a private bathroom), depending on where you chose to live and how far from the MRT you are. I strongly encourage you to find your own private space with a lock that only you have keys for (and ALWAYS keep the door locked).
Foreigners tend to come to Taiwan with the belief that everyone here is nice and no one will harm you. Programs heads tend to enforce upon foreigners how much safer Taiwan is than other places. However, in my experience, while walking around in the middle of the night isn’t a threat to your life the way it might be in NYC, it doesn’t mean your housemates won’t turn on you within a few days of living with you. No matter how nice they might seem at first, once the “honeymoon period” is over, you’re going to find things going missing, the shared kitchen/bathroom turning into a nasty science experiment, and, in one case for me, the landlord replacing the locks and not letting me access my stuff after I called him out for allowing a family of 6 with two babies to rent the 8 ping room next to me. (The landlord himself was subletting, which ain’t legal)
That’s the surface of my horror stories. I really don’t feel like going into details. If you’re hoping to meet people, start off at a hostel for a few days before moving into a private room in an AirBnB. There are plenty of hostels in the NTU/NTNU area that don’t really care if you chill out in their lounges doing your homework after you’ve moved out. (This makes me think twice about the security of your belongings and yourself, but it’s also a great way to make friends)