Unpopular opinion maybe, but I think she is partially in the right.
The fact that you went out of your way, she is wealthy, she has a bad attitude etc is not relevant in my opinion.
The only thing that is relevant is that you did the job without agreeing the price with her first. That means it is well within her rights to pay whatever she deems reasonable.
It isn't unprofessional for you to accept £50.
It would be unprofessional for you to start bringing up the fact that you're only a small business, and have a baby etc. Totally irrelevant and not her problem.
So, just accept the £50 with a "I guess I really should've had the price agreed before I did it, oh well that's a lesson learned for me"