If my LCD was blacking out, I'd just be tempted to buy something like this off ebay for sub-$10, and separate the LED from the electronics and stuff it all in the spaces within the original temp gauge. If you look harder you can get ones that are "auto" dimmable (input from headlight switch kinda thing).
0 36"Blue LED Digital Car Thermometer Temperature Meter 55 125°C DS18B20 Sensor | eBay