So sorry to hear of your struggles with your mare. Breeding can be so heartbreaking (horses in general, I suppose) but this sounds like a great opportunity and I hope it works out for you!
I don’t have any specific experiences to provide, so just “thinking out loud” here with some ideas to consider. Some of these, would be more on the mare owner to know and consider, but would be the kind of things that I would be thinking about if I were to offer my mare for this sort of situation.
Is this mare competing? If so, that would definitely increase the price, since it is summer show season.
Is the mare close to you, or would she have to travel a long way? Does she need to be at a vet clinic for the whole time? Does the mare travel well and settle in at new places easily? Will having the foal add value to the mare in the future (e.g. will she be eligible for “Elite” or similar status with her breed registry or potential sale as a broodmare with a production record in the future)? Is the mare one that can take a month or so off pretty easily and come right back? What happens if she doesn’t catch the first time?
As a possible starting place, it seems like annual leases are 30-40% of purchase price these days, and you could divide that by 12 to get a starting estimate for a monthly lease, although I would be prepared to pay a bit more than 1/12th, and add insurance and shipping on top. For example, if the mare was worth $50,000, 35% of her value would be $17,500, divided by 12 would give you a value of just under $1500.
Then you can go from there and increase or decrease depending on whether or not this is considered less risk because she isn’t being ridden and shown by someone else like a traditional lease, if having time off is going to interrupt her training or show schedule, how far she has to travel, how much stress this will be on her, how much of an inconvenience this is or isn’t for the owner, if there is any value added to the mare here, etc etc.
I might be way off, I have no idea what would be standard for this, but that is at least how I would start approaching it!