90k is a lot for Taipei. You can easily find an apartment that will suit your needs with that.
Don’t fall for serviced apartments, they are much more expensive than ordinary apartments.
Have a Taiwanese look for you if you can’t speak Mandarin.
I found a nice 2br place in Shipai, in a new highrise building about 3 minutes from the MRT.
3-month lease, 45k per month and no deposit.
60k is reasonable for a 2br in Daan, maybe as much as 70k for Xinyi if you want a 3br.
I’d advise against Xinyi, it’s not really worth it. Getting around is awkward since it’s at the end of the MRT.
Tianmu is another expat favourite, and is fantastic if you have a car - lots of restaurants and businesses catering to expats. Most people will speak English, and landlords in the area will be more familiar with the needs of expats. The only downside is the lack of MRT access (most would complain about higher rents, but it’s quite reasonable as a Londoner, and still cheaper than Daan and Xinyi).
Zhongshan is decent if you can find a nice apartment in the area. Great MRT access, great location, plenty of restaurants. I had trouble finding any attractive apartments there though.
Daan is good, lots of nice restaurants and nicer buildings in general. MRT access can vary, some parts of Daan can be ~10 minutes walk from an MRT.