Ideally - Daan District, 25-30 day stay, 400-500k budget.
Flexible on most other aspects.
Edit: never mind I misunderstood what 月子中心 means
I don’t live in Taipei, but if you want to limit your search to Daan district, there are actually only 4 or 5 options.
There are some in near-by districts sucha s Zhongzheng and Zhongshan.
Plenty of them have rooms between 600K to 800K. However, I read on some reviews that some smaller rooms can be as low as 120K.
The prices for these places are extremely non-transparent, sometimes even when you call them, so you might just have to schedule a visit to get a quote.
That is of course unless someone with actual experience chimes in.
I used this place and it was very good. 4.9 stars on Google reviews.
regular rooms are 200-240k but they have VIP suites if you are into that.
英倫產後護理之家 | 台北產後護理之家、台北月子中心首選 (yinglunbaby.com.tw)
They are in Daan district: No. 230號, Section 2, Keelung Rd, Da’an District, Taipei City, 10675
Speaking from experience, I think you can find for a lot less, probably <250K. Dress down and go talk to a few places.
Also, the mom will probably get sick and tired from being there around 3 weeks. I very much doubt you will be adding time.