He does kind of have a point though. It's kind of like asking for a cheap coffee. Just go to 7-11!
You could at least give a budget.
There are a lot of cheap Giant, Merida and KHS bikes in the 3000-5000 range. They are city bikes with (as far as I can tell) ultra cheap mountain bike components. You can look on their website.
There are lots of other cheap-only local brands like Irland, Jelum, TongMing etc. You can see them everywhere.
If you want something you could ride around the island then maybe a cheap hybrid would be a better option. You should be able to pick up something like the Giant Roam 2 for around 12,000. I rent one of those sometimes when I go to Danshui and they seem perfectly tolerable.
Not sure how easy it is to find a larger sized cheap bike. The standard for cheap student/foreign worker bikes here seems to be dual-suspension kids bikes
. Anyway, just go to any bike shop and ask the laoban.