Can I repair a Texas instruments TI-5135

I don't know if you can even get a new print head anymore, that calculator is over 30 years old! You may be able to get by with cleaning it off though. There's some tutorials on YouTube, one video I found used 99.5% isopropyl alcohol and a tooth brush: https://youtu.be/QWD5iwWl_MI?t=14m34s Hopefully your calculator prints the same way.

Unfortunately, I have no idea how to take it apart nor how to get to the head. However, those old machines normally are reasonably easy to disassemble so you can probably figure it out.

Sorry I couldn’t help much, however if you run into any problems I’ll still try to help! Just comment to this answer and please share some pictures if you can; pictures can really help diagnose what’s wrong or what needs to be done.