If there is a 12V Lead-Acid battery in there, it is likely shot. The resting voltage on any 12V vehicle ( with a lightly-loaded, fully charged, healthy lead-acid battery should be about 12.65V, higher if the vehicle was recently run and the surface charge on the battery hasn't yet dissipated. With the charging system on-line, you should see about 14.2V or a bit higher.
Is there a big load on the battery when you turn on the key, like the headlight? That might pull the battery voltage down a bit, but 11V looks worrisome...