Worst case scenario, you can fly to Taiwan with your family and enter the country without a visa - with an American passport you will get a 30 days visa exempt stay stamped into your passport. Then you can stay in Taiwan for 30 days - then fly to Hong Kong - get off the plane - and fly back to Taiwan and get another 30 days stamped in your passport, and so on.
I did that for more than 2 years. The flights are short usually 1 hr 15 mins, depending on the airline - EVA are cheap and Cathay Pacific are ok too.
But if you can explain your situation RE your family staying in Taiwan and your need to join them on a VISITORS visa maybe you will be lucky and get a 60 Days EXTENDABLE VISTOR visa. (they can even issue a 90 days visa, twice extendable by 90 days but that is only in the regulations - in practice it almost unheard of these days) (I think).
That will allow you to enter Taiwan for an initial 60 days, but then apply (whilst still in Taiwan) for an extension of a further 60 days - which you can do twice, giving you a total stay of 180 days on the visa. Would you be applying to a Taiwan office in America for your visa? I don’t have any experience with the American taiwan offices (TECOs) but I would suggest, if you will apply in Asia, that you stear clear of the Hong Kong office for Visa appliactions and go to somewhere more understanding and friendly like Osaka or Macau.
However you can never be sure of a pleasant experience at any office if you catch them on an off day - just some have a lot more off days than others.
It will be a major plus to your case, if applying for an extendable type of visitor visa to be able to show documentation of sufficient funds to support your stay in Taiwan (3 months of Bank statements / savings, etc.) remember if you want them to give you a 60 days visitor visa - that effectively gives you 180 days in Taiwan, they will want to see enough funds to support that length of stay. Some posters on the forum have suggested that about 2000 USD per month was enough for their application.
If you can combine your:
reason no.1 joining children and partner staying in Taiwan.
reason no.2 sufficent funds to prove you won’t become a burden on their society, or need deporting, for the duration of stay you are asking for. and
No. 3. The choice of a more sympathetic and friendly TECO office to submit your application to.
Then maybe you have a good chance of getting somewhere.
You could try explaining your circumstances and asking about your options to someone on the “Information For Foreigners Hotline” (belonging to the Taiwan National Immigration Agency).
(But I wouldn’t ask them about which TECO office is friendly or unfriendly, this forum is probably a better place to ask people about their own experiences with regards that.)
Another alternative is enrolling on a government approved Chinese language course at an approved school and getting a student visa that way, you will need to show good attendance at the school in order to keep extending your visa.