Brand new to this forum but I use them to repair every vehicle i have and find them to be the #1 resource. My niece has a 09 LR2 that has on occasion not started, replaced battery because when i was called to help indicated 12.1 , dead flat. This has happened a few times and pulling the battery to charge supposedly resolved the issue, not this time. I have been told if the key fob battery goes dead it will not start but self test initializes and that there is a 15A fuse for the circuit but fuses don't heal. It seems to be recurring but long enough between incidents to make it confusing. There is a light relay that clicks and nothing else, Is this common the the vehicle? There is an R7 relay that has a key picture but the fuse/relay locator on the cover is enough different that certainty is lacking.

 

The keyfob battery has nothing to do with starting, it only controls the actions of the buttons. It sounds like your starter has gone south. Symptoms of that at first is the battery getting drained, and eventually the starter just dies. R&R is a fairly easy DIY job and the starters can be found on eBay.

 

Yeah, I'd have a hard look at the starter, replaced mine with a new one sold on Amazon. And, while you are at it check your ground wires.
My failing starter symptoms were just slightly slow cranks, once replaced, the car starts immediately, zero lag. It's easy enough to get to.