I searched on united.com for availability, it gives you a calendar view.

JFK to TPE had availability in business on 22nd Nov

It might be a good idea to consider Chicago which has two dates available...
13th and 22nd Nov: ORD-TPE

As for flight to DPS, seems like that might limit you further. You may wish to consider alternative destinations than DPS as Eva fly once daily I think, e.g. go to Singapore or Jakarta and then take a cheap flight to DPS.

If you're completely flexible you can fly on 22nd JFK-TPE-SIN which is a pretty good situation and then do whatever for SIN-DPS.