You can look online or get help from Taiwanese partner/friends. Need to check the Chinese websites.
The other way is to look for the realtors in that area, usually they have a little outlet near the bottom of these large apt buildings. Not all of them do rentals, but some do and a minority specialise in rentals. Go in and tell them your budget, how many rooms, ping, how old etc. They can usually show a few right there and then. Actually you have discovered the secret of Taiwan, it is extremely good value now compared to most Asian countries…you can live in nice new places for small change, your deal of 33k/mth for 50ping in Zhubei is about right for Zhubei.
There are beautiful new apts going for 3-4 million NTD in Taichung, Taipei you are looking at double to triple that, it is even more expensive in dumps like XinZhuang than nice places in Taichung. The rent for a new apt fitted out in Taichung with swimming pool etc. is 20k/mth starting, impossible to find in Taipei City. The rent to purchase ratio is higher than Zhubei, I think that it is a good indicator that purchasing a house in Taichung is more reasonable. But I don’t much about property.
The problem for most folks with living in Taichung is that salaries are much lower in general and they don’t have the jobs that can be found in NeiHu, Nangang, Xinchu Science Park . Taipei commands a higher price because salaries are higher and retired folks, govt agencies and people who think think they have class like to live there