How to replace AC/DC adapter but keep old connector
Chop and solder. Of course, if the current adapter is dead, it can be difficult to establish correct polarity - but I think I see that labeled nicely on the present adapter (it's not always, though it should be.)

For a tidy job, split the cords and cut the + and - wires at different lengths (so the joints are slightly offset, rather than being at the same point) slip a large heat shrink over the whole cable and one or two smaller heat shrinks over the individual wires, moving them far enough from the joint not to be shrunk when soldering. If the offset is large enough you can skip the inner ones entirely and just have the outer one (the offset keeps the exposed joints from touching each other when held by the outer heat shrink.)