I would look at it from your tech's eyes and from your customer's eyes.
She brought the car in and she got it back with something she had to take care of. If you take care of her, even if she is willing or says she is willing to take care of it on her own, then she will be your customer for life. If she has to pull teeth to get you to fix it then she isn't coming back. Making her happy in this case will cause her to trust you more than if nothing had ever happened.
Your tech lied to you because he doesn't know how to fix it. He doesn't want to be responsible because he isn't. Being lied to and letting him get away with it sets a precedent to permit that kind of relationship/behavior in the future, not just with him but with all your techs. Not good.
I am not perfect and this is probably not the best solution but here is what I would do:
1. I would fix the customer's car myself because I don't want anything to be at all wrong to cause her to come in again; or worse to not come in again because she feels sheepish to complain again. If I felt so to do I might repair whatever was causing it to not work in the first place. It may cost a lot of money but she may then offer to pay, but at a minimum she will be referring anyone she can to your shop.
2. I would tell the tech that I value him and his effort to make my business a success, but that trust is of utmost importance in the process. I would tell him that I hope that he would feel comfortable telling me that he broke it and that he is uncomfortable fixing it if he thinks he isn't capable. Then I would state to him that I know that he lied to me and that if that happens again that his employment would be terminated; this isn't up for argument this is a statement. If he tries to argue that he didn't lie then he still is not trustworthy and I would fire him right then. Everybody makes mistakes, but that doesn't make it acceptable because someone has to pay for it.
You are the shop owner and the tech is the employee, you bear the risks of running the business; however I see the viewpoint of others that the tech should be the one to pay for it and I would agree if I trusted him to do it properly; but based on the fact that he may only be a tire tech and that you are a risk bearer yourself and that I in your shoes am thinking about my future relationship with a customer who may have lost trust in me (not my tech), I would have done as I have stated above.
Sounds like you have things figured out.