Charles touched on it ... you need a 2.1mm x 5.5mm DC connector wired center positive for the scope side connection. How the other end connects to your batteries will depend on the particular battery product but most will have the automotive cigarette lighter style socket.
There are also 2.5mm x 5.5mm connectors that look exactly like the 2.1mm versions and fit into the socket on the scope but they will often introduce intermittent power connections.
The most important thing to verify though is that everything is wired tip positive (which is the way 99% of the commercial products are wired).
If you successfully clean up the battery compartment (it should clean up without actual damage) you can add alkaline AA batteries and use them as a backup to the external power connection. With the internal batteries any intermittent power connection won't cause the mount to reset (and force you to go through another initial alignment). The problem with soley relying on the internal batteries is that they get drained very very quickly and with low voltage the mount starts to act very strange. Just don't forget and leave these batteries in the scope for long periods.
So basically your choices come down to 3 general types. A small 12v Lithium power pack with a built in cable (with that 2.1mm connector), or a jump start style lead acid 12v battery (heavier and larger than the Lithiums) plus an external DC power cable (2.1mm x 5.5mm on one end , cigarette lighter style on the other), or if you have a 110v outlet and extension cord handy an inexpensive AC to 12v 2-3 amp (and more amperage is fine too) regulated power supply (also with a built in 2.1mm connector).