First this is a very old system! I had one which I loved until it died. This series uses CCFL (Color Corrected Florescent Lamps) [guide|464]. These tiny lamps are no longer being made and finding replacements is extremely hard. In addition the Inverters tend to wear-out which are also hard to find that still have any life left in them (pig in a poke) [guide|1442].
It does appear you have a workable solution presently using the external display.
Maybe its time to shed a tear as you move on to a newer system. You don’t need to get the very newest model. I’m going to assuming you have a lot of older OS-X apps running under Lion (OS-X 10.7.5) and stuff that won’t work on the newer macOS Sierra (10.12.x). So finding one of the last OS-X systems like El Capitan (OS-X 10.11.x) would be one possibility. Even still you really don’t want to do anything on the internet as the older OS-X versions which don’t have support any more have security issues! Anything on your system is at risk.
If you don’t have anything overly important then getting a newer model that runs the latest macOS would be a good investment. As an example a [https://everymac.com/systems/apple/macbook_pro/specs/macbook-pro-core-i7-2.9-13-mid-2012-unibody-usb3-specs.html|2012 13” MacBook Pro Unibody].