Will be in Beijing for work for a month and my mother reminded me that I have no visited my relatives in Taiwan since I was a little kid.

Coworker of mine said it is about 7 hours door to door for PEK to Taiwan. (I assume its Taipei airport)

What are some possible times for:
1. Leaving LATE friday (I assume it will have to be after work)
2. Leaving EARLY saturday
3. Flying back LATE sunday (so I can go back to work Monday morning)

What are some airlines I should look into?
I am sure there are small airlines that are not on expedia.

Thanks!