You handled it well. You took full responsibility and took appropriate steps to address the issue. That’s the best thing you can do in situations like this. Keep the customer well informed, don’t try to hide any issues from them, and give them an action plan on how you’re going to resolve the issue. That’s the best possible path you can take when issues appear!
Only thing I’d suggest to to make the customer aware that repairs don’t always go to plan, and there’s a small, but non-zero chance, that further issues may arise after repairs. They shouldn’t worry too much, and you’d be happy to discuss the issue further with them if anything does come up. Customers need to be aware of, and accept, that small risk with any electronic repair, and if they don’t, you’re going to be in serious trouble if something does come up.
There will always be an issue now and then when fixing anything. Repairs don’t always go right, people make mistakes, and sometimes, luck just isn’t on your side. As long as you continue this level of respect and clarity, I think you’ll do very well.
Good luck with your new repair business!