given your tight budget I agree it might be a little but of a struggle.

I would look for an apartment on Airbnb or a similar service to save some money.

I think there's enough to do in Taipei for that amount of time. And public transport is extensive so you can even get a bit out of the hustle and bustle of downtown quite easily.

I liked the tea house area reached by the Maokong gondolas south of town for some fresh air and nice views. As well as a good walk.

And Tamsui/Danshui was also nice for an afternoon/evening.