As best as I understand, the OP's DH wasn't around when SIL jumped in. Then, she tried to take the credit for the success of the entire project.
There is a whole lot of baggage here, as made obvious by the fact that FIL didn't shut SIL down when she started up her act.
I have mixed feelings about the OP's DD writing the SIL a thank you note. Yes, the SIL did help, but in such a way that it hurt the FIL's feelings. On the other hand, the FIL could have shut SIL down in a heartbeat.
The OP says that the SIL often acts this way to the entire family, so basically, she seems to be getting a pass on her bad behavior. She seems to be the "Oh, she's just that way" relative.
I'm hesitant about the OP's daughter writing a thank you note because it enables and e ncourages her to act badly. On the other hand, this may not be the hill the OP wants to die on--it sounds like SIL will give more opportunities in the future to make clear what is nice and what is not.
The only thing I can think of that would be e-hell approved would be to send FIL a video, photo, or taped message of DD saying thank you, etc. SIL gets a short one sentence thank note, along the lines of "Thank you for helping FIL with the paint."
There has got to be a way for a smart person to be polite and yet not reward the bad and scene-stealing behavior of the SIL.