SIL is POA but not helping us take care of my MIL.

Is there an Advanced Directive? Who has POA for healthcare? You said SIL has financial. If SIL has all these things, it's time to dump all related matters in her lap. You've done enough. You've got enough on your plate.

Depending on where you live, your home is considered your MIL’s residence and you’d have to go through a court process if she doesn’t want to leave. Even if she hasn’t paid anything to live there, she could still be considered a tenant. Check your state laws on this. 

You can’t force another person to take care of her regardless if it’s her own daughter. Sounds good on paper to essentially knock on SIL’s door and leave her mother on the porch for her to deal with but be careful. It could turn out the way you want but it could be a legal nightmare.

I’d contact your local Human Services Agency on Monday and get the ball rolling on having MIL evaluated for placement somewhere. Give them SIL's name, address, and phone number and tell them she is POA and to contact her in regard to everything. Once SIL has some responsibility dropped on her, and has to answer to someone with authority, I’m betting she’ll get off her behind. 

Take MIL to the SIL house . Tell her you and your husband have an appointment and don’t return to pick her up .
Have her belongings delivered to SIL house .

SIL needs to stop stalling and hire a RE Sales Agent. She has no excuse.

You have no power in this situation other than to say "no" and keep saying it. They way you say it emphatically is to drop off your MIL at SIL's house. Don't tell her why you're coming, just make sure she's home first. Where is your husband in all of this? Hopefully he has a spine...

The house will need to be sold at fair market value in case Medicaid is ever needed. Tell SIL to take the next offer that is FMV. Or again, your returning Mom home and she can supply her care or hire someone. Mom can be placed now and go on Medicaid if needed. A home is an exempt asset while Mom is alive.

POA is just for financial and Medical. The assigned person is not obligated to care for the principle or be at their beck and call. But, they can hire someone to do the care. Clean the house, mow the lawn, all out of the principles money. Its the POAs responsibility to place the principle when needed.

Why did you take the woman in when you and DH are both disabled, I don't understand? Alzheimer's is a very difficult issue to deal with and only worsens with time, as you see now that you're mentally burned out after 4 months. Was the goal to sell the house to place MIL in Memory Care Assisted Living?

Your option is to drop MIL off on SILs front doorstep, lock stock and barrel, citing your inability to care for her any longer. SIL dragging her feet with the home sale has now caused you and DH extreme anxiety and health issues from caring for MIL while being ignored by SIL.

I'll bet that home will be sold immediately and MIL will be placed in the blink of an eye. Empathy is earned thru scar tissue, and it's now SILs turn to see what it feels like to earn her title.