Doesn't matter which lead you put on the outer and inner connections, it should read something, just that if you have it backwards it will read a -ve voltage.
Mostly the inner is +ve, but not always.
Does it have the voltage/current rating on the adapter?
It should not be hard to replace, most now are switching supplies and precise output, if yours is an older one, it could be a linear supply, if so and you can open it up somehow, probably can be fixed.
Al.
CNC, Mechatronics Integration and Custom Machine Design
“Logic will get you from A to B. Imagination will take you everywhere.”
Albert E.