Doi Saket is the outskirts of the Chiang Mai Metro area. Note, though you mention Doi Saket hot springs - you might want to look at a map, Doi Saket ranges from the Chiang Rai border to the Outer Ring Road.
If you can compromise on nature, take a look at the southern parts of Chang Klan and Haiya - close to town but off the major roads they're quiet and mostly Thai (the northern parts are highly touristy).
Some people love Santitham - again, there are quiet parts and yet you're very close everything.
A little further out Fa Ham might be worth a look as well - close to Central Festival but outside the Superhighway it's fairly quiet.