Serviced apartments can be rented on a nightly basis.
That's the case for all the ones that I recommended in my earlier post and others that you'll find on hotel booking sites e.g. sawadee, asiatravel.
Sometimes they have a minimum stay but usually that's only 2 or 3 nights and even then, that's for peak periods.

Just got back from Bangkok - stayed at two different apartments (both in same area - just off South Sathorn Rd., handy to Silom).
At the start of our trip we stayed at Somerset Park Suan Plu for two nights and at the end, we stayed at Pantip Court - both similar price and similar facilities.
Suan Plu is newer and fancier while Pantip has lovely garden/pool area and apartments are bigger. Both around 3000 baht for a one bedroom apartment.
We also visited friends at Somerset Lake Point - 2 bedroom/2 bathroom apartment around 3500baht a night. This could be a very good option for you - cheaper than the others I've mentioned but similar facilities.

You don't say when you're travelling, but prices vary according to season... July/August usually cheaper rates than this time of year.