Taiwan is a fine place to visit! It's really something of an undiscovered gem so you'll have all that natural beauty and awesome culture almost to yourself (and hte domestic or Japanese tourists, but Westerners hardly at all).
It's very safe for solo travelers and fine for females. Seriously - crime here is minimal and the Taiwanese are laid-back, friendly, welcoming and hospitable. This is one of the easiest countries to be in that I've ever experienced! I've lived here for 3 years, much of it solo (I'm female, too) though my fiance is here now, too. He liked it so much when he visited that he moved out here. English is widely spoken in Taipei but not so much in other areas. Never fear - people are friendly and will try to help you, even if that means going out to fetch the one guy who speaks English on the block. Get a phrasebook with characters you can point to and you'll be fine.
US $40 a day is fine in most parts, but in Taipei I'd recommend US $60 a day as it really is more expensive.
