Sounds like the battery itself is toast.. There are basically 4 battery failure modes. One is very rare
Shorted cell: One or more cell shorts out, this means low voltages which is what you are seeing
High Resistance: The voltage is there, but if you draw power off the thing it drops like a rock, returning when you remove power, also will not charge very fast.
Low Capacity: Basically it no longer holds any power to speak of. Goes from FULL to Empty faster than I can type Full to Empty.
And the rare mode is "Exploaded" (We say Explosion but it's more a POP than anythign, Major mess you need to wash the battery compartment with Soda Water (Garden sprayer handy) and all over the compartment.. Then you need to flush with clear water, then replace battery.
In all cases it's time for a new battery.
What ran it down? I am assuming you are not on shore power.
How much battery do you have? My RV came with a pair of GC-2 Golf Car 6 Volt in series (And by the way this is the type of battery I recommend if you can house them, best Bang for your BUCK epically if you are a members of SAMS or Costco).
This is 220 amp hours (now upgraded) and was able to run the Furnace and TV overnight when new.. As they aged the amount of time before I got a low voltage alarm went down to about an hour. (Disabling the inverter, and thus the TV improves run time)
I have since upgraded those batteries (I mean it was time, 9 years) to 230 amp hour replacements and added ... Well I have about 500 Amp hours.. I can run all night again, TV, Radio, and Furnace.
But from the 11 volts peak, Sounds like shorted cells in battery.