I'm going to Taipei soon and found out about a left baggage storage facility about 100-200 metres away from the main Taipei train station: http://crazy_azn1.tripod.com/taiwan/id32.html

NT$ 17 for up to 10 days? That seems awesome. The negative part is it closes at 8pm. If cost is not a problem, see this as an alternative should the lockers run out.

If you read Chinese, here is a blog that provides directions+pictures from Taipei station: http://yumeko.gogoblog.tw/blogview.php?blogid=441 Otherwise, just a quick translation:
1. Head to East-3 exit (apparently exit with lots of coaches outside)
2. Cross the road
3. Walk straight along the pedestrian walkway
4. You will end up at another junction, with the walkway turning towards the left and no walkway on the other side of road. Follow this pedestrian walkway and turn left.
5. Eventually you'll see the baggage storage facility on the opposite side of road.
6. Head in and someone will assist you.

Closes at 8pm, might be a little early for you to head to the airport.


EDIT: Did not realise this post dates back to 2010.