you want to go jiu fen & taipei from taichung? for how many person?
I would suggest you take the high speed train from taichung to taipei... it's cheaper
coz a normal 4-seater with driver will cost you NT5000 to 5500 (which is over $200!) and 2 1/2 hr!
a high speed train ticket is only S$20+/pax and more comfortable and faster. (about 30 mins).
then once you reach jiufen just take a bus or normal train to go (less than $8 / pax), google the internet and you can easily find details on how to go jiufen.
if you really want a driver... i suggest you get one driver in taipei and ask him to pick you up from taipei station where you alight from high speed train.
for a taipei driver to drive you back & fore jiufen may be around NT2000,
if you just take cab on the spot may be even cheaper! There are lot of cabs queuing at jiufen main road (as seldom people will take cab).
you can ask your driver to also take you to visit shifen, keelung night market, yeliu for whole day at NT4000.