If it's reinforced security glass, then how could you break it with a brush? Sounds odd!
If it's not reinforced security glass, which I doubt (on a first floor DW home!) then just get some alternative quotes to replace it for normal glass and get it sorted.
Your reputation is at stake here! I am guessing you never did it, as your cleaner seems honest. I am also guessing that it's not that expensive to replace, so get some cheaper quotes and just pay for it.
I’ve passed it to the insurance company. I think that’s reasonable. Let them sort it out. I’m sure they’ll pay out.
Getting additional quotes will cause the customer inconvenience and create additional hassle for me. It’s what the insurance is for. It distances me from the problem and is certainly more professional than either a DIY attempt or getting various people in to quote.
I don’t think it’s possible to have broken it with a brush and it was cold water.
There's no way you broke that window and I would be surprised if the customer didn't know that, you should never reward dishonesty.