TDR to detect broken LED
No, a Time Domain Reflectometer won't help in this situation. A TDR sends a pulse down a cable and analyzes the reflection. The pulse will never get past the series LEDs. Well, maybe some might get past, but not nearly enough.
I have used a non-contact voltage tester to find breaks in line-powered LED strings. You may be able to inject AC into your string and do the same. You would need to find a safe voltage that was still high enough for the tester to detect.
Another option is to pierce the insulation with a very thin pin/needle. Use a binary search to find the issue. Recommended for low voltage circuits only.