When you apply for a Visitors Visa you will notice a BIG stamp with two dates. This is the time during which the visa is VALID.
When you arrive in the country you will receive a SMALL round stamp with a date on it, this is the date from which your 60 days start.
Last year I made the mistake of looking at the BIG date stamp, and suddenly found that I had overstayed by a few days. I therefore left the country Immediately (within 7 days) and had to pay a NT$10 000 fee at the airport.
I then went to Singapore and receive a new non-extendable Visitors Visa with no problems. I did ring them first (or rather I asked a Taiwanese friend to do so) and asked them if me overstaying would be a problem. It was NOT.
However, if you overstay longer than (I think it is 10 days) it is a more serious matter. You have to pay a LARGER fine, and I am not sure if you get any special stamps put into your passport.
Going by the dates you have given it looks like you have overstayed for longer than 10 days.
I suggest you ring the Singapore office and talk to them first. Ask to speak to the director / supervisor, NOT the 'counter girls'.
Depending on his answer you may have to go back to your home country instead.
I am sorry I can not give a more specific answer.