So, failing backlight LEDs seems to be common issue with these Keithley 2110. I've seen it on a couple of units, but they've been on permanently for >5 years, so it's wear-and-tear rather than a fault IMO.
The light comes from two small PCBs mounted at each end of the plastic block "backlight diffuser". See attached photo. One of these PCBs is 'surface mounted' on the display PCB. The other is through hole and is trickier to remove - I just snipped the legs going into the display PCB. There are some clear plastic tabs that need snipped out too.
Anyway, replace the three SM0603 LEDs on each backlight PCB, refit and its job done. Quite easy.
Make sure the PCBs go back on at their original location, i.e. putting them back on the wrong side of the LCD will reverse polarity, won't work.