Taiwan tours advice

My husband and I will be in Taiwan from 4th - 19th Dec and our friend has graciously offered us the stay of his home in Taipei Beitou district.
We plan to visit many other touristy places (hualien, kenting, sun moon lake, etc) so I wonder if it would be possible to make Beitou our 'home base' and do day trips to these attractions. Or would we be better off staying in hotels in some places?
We have not fully planned our itinerary since we prefer relaxing tours rather than jam-packing multiple locations in a day. Is Dec peak period and would hotels/min su be easy to book on short notice?
Thank you in advance for any inputs.