Mother is on Medicaid living with us. If selling home, we would like to recoup labor costs to repair home in 2010 when purchased, how do we?

A good place to start is "Medicaid's Power To Recoup Benefits Paid: Estate Recovery and Liens. I do estate planning and I have run into this situation many times. Medicaid is like a bull dog you can't shake for repayment of benefits paid. Not being on the title/deed just complicates the situation, although, they can still come after the paid funds if you were. They are very, very difficult to deal with.

You cannot recoup labor costs or any costs without a written contract in any situation. It seems that it is too late to say this now, but you should have had an assessment done and a contract written up by an estate atty.
That being said, Medicaid will recoup all of their losses. If your mother sells now, you will be left out of the equation. You may get some options or relief if you consult with an atty now but I don't think you can recoup monies from the past without written contracts. If that were the case, everyone would be doing it. If you do not want to sell given the likelihood that you will not recoup costs, you can apply to be an unpaid family caregiver. There are programs that pay family members that provide 40+ hours of care giving, but it depends on your state. I would check with your state/DSHS or county.
I'm not sure what your objective is, if it is to receive money from her home sale or to get her into a facility. I'm also confused on how you co-signed on a mortgage but are not on title. You really need to pay an atty to look at all of this.

A little late, you have already let the horse out of the barn. It is her home, you live with her, she doesn't live with you. Medicaid will take any profits from the sale, before you stepped into this arrangement you should have done the research and without a legal written document you are basically out of luck.

This is her home. Unless you can prove you have a vested interest with contemporaneous documents, Medicaid will consider the entire house hers alone. They will seek 100% reimbursement from the house sale.

if you sell the house now, while she is still living, Medicaid will stop all payments until she can prove that 100% of the proceeds have been unused to pay for care (spend down).

I would check with Medicaid, but I would say that probably not unless you sent a bill 9 years ago and placed a lien on the house at the time. Do you have timesheets and records for what was done?
Do you have proof that they received the bill, such as return receipt requested from the USPS? Was there a written agreement that says you would be reimbursed from the sale of the house?

I recommend that you contact Medicaid and let them help you understand what they require and how your state works.

Sometimes we just have to chalk things up to education unless we want to take on 24/7 care until the penalty period is over. Assuming you didn't live in the house as a caregiver, is that correct? How long has mom lived with you?

From what I’ve read here, when someone goes on Medicaid, a lien is placed on their home. When the house is sold, at fair market value, the proceeds go to pay for her care. Recouping costs is risky at best because you’d probably have to research the cost of each task you performed, that is, you couldn’t ask $100 for changing a light bulb. Any money you received could be considered gifting and she could lose her benefits.

Can you clarify a few points? When you say "20K of their money", who is "their?" Whose money? Also when you say your Mother is on Medicaid, what kind of Medicaid is she on? For medical care or for long term care services?

Regardless of Medicaid's involvement, I doubt you would be able to recoup the value of your services unless there was a written agreement to that effect. If Medicaid has a claim to the proceeds of the house, it seems even more unlikely, but it seems unlikely to me anyway.