How can I do touch-ups and repairs on my gel nails at home?

unless you have the proper equipment, you can't.

I agree, you cannot do it without the proper equipment and supplies. It will cost you several hundreds of dollars at the very least for professional products. And a good UV lamp will cost you from $150-$200 all by itself. But, if you don't have a license to do nails, you cannot buy professional products. So, back to square one now. Let's just say that you were able to get the supplies, equipment, and all the products that you need - what will you do now that you have it all? You need to know about product chemistry, nail anatomy, the proper methods of application, cure times, which products to use - and when to use them, and so much more. My advice to you is to leave this sort of thing to a professional. It is much more difficult to do than it looks. I do both acrylic and gel nails, and gels were not easy to learn - that's for sure. I have since mastered them, though it took a LOT of patience, and a LOT of practice to get to the point where I am now.