If your Visa Exempt expires on Feb. 27th, you need to leave the country and come back. Go to Hong Kong. Use a travel agent here in Taipei, should be able to get you a RT ticket for around $6,000 - $9,000 NT. (Expedia or other websites are not the way to go here, you will always
pay more with the internet).
With regards to Visa Exempt --> ARC: this is how it works.
1) Find a job that will sponsor you on an ARC + NHI. Sign the contract. The government requires a minimum 14 hr/week contract to qualify for an ARC.
2) Your employer will apply for your work permit. To do this they will need from you: current Health Check(<3 months old), 8 ID photos, a copy of your passport, tax receipt from the previous year (if you worked in Taiwan last year).Your work permit should arrive 2 weeks later at your employers office.
3) Get a Visitor Visa (This can be done by you or your agent): Bring your work permit and your passport to the 2nd floor of the MAC Building
(not the Immigration Office) and fill out an application requesting a Visitor's Visa. That's right, if you have a work permit in hand, you can apply for a Taiwan Visitor's Visa while IN TAIWAN!
Your Visa will take anywhere from 1 day to 5 days to be issued. Once issued you will now be in Taiwan on a 60 day visa and shouldn't have to make any more visa runs!
4) You or your agent can now go to the Immigration Office
and apply for your ARC.
My employer did all of this for me (acting as my agent). With the health check included, it will cost around $5000NT after all is said and done. From the date your employer applies for your work permit until the day the MOFA issues you a Visitor Visa is around 21 days.