If you decide to replace the CCFL's with new, replace all of them, not just the dead one. The others will likely be near the end of their life as well.
How is Brightness and Sleep (switching off backlight) handled with the replacement LED strips? There doesn't appear to be any documentation mentioned in the ads for the LEDs, although they seem to have some sort of connections for Enable and Brightness in addition to their power leads.
The one I did (a 22 inch BenQ) had 4 wires as input to the converter board. Vin: 12-30V DC, ENA: enable, DIM: brightness, and Ground. It's up to you to find ENA and DIM signals on your monitors logic board, or you can leave them unconnected for full brightness.
As I recall, I got my LED kit from Aliexpress for under $4. Check the feedback and pick a vendor that packages them properly, as the LED strips are easily broken if bent. Mine arrived inside a 2-foot piece of cheap plastic conduit for protection. And I tested them with a lab supply before I confirmed delivery.