I really feel for you - I had a similar issue with delays getting a letter (that I paid £30 for) from my GP. You won’t have to pay for the patient care summary though - it should be free and it’s a request for your personal information.
It’s unacceptable that they won’t provide you with a summary because of a broken printer. 4-6 weeks is also ridiculous.
I resolved my issue by calling my GP and dropping in every day to see if they had progressed. Most of the staff were on my side but the manager was a ‘computer says no’ person who seemed to not have a clue what I was going through.
I think by having other staff on my side, they were able to go around her and help me push it through a bit quicker.
You can call Visa Medicals in London and they might be able to guide you on what to do. They were really helpful for me. It’s better to give them a heads up about what you might have missing on the appointment date - you can potentially email it through to them afterwards.
Other than that, I had to let it go a bit and resolve that my life was in this ignorant person’s hands and worrying about it was only hurting me. It will be alright in the end.
If worse comes to worse, are you in a position where you could change the visa medical appointment date? I changed mine a couple of times with no negative consequences for my application.