Correct size should be 3.5 x 1.35 mm if yours is the same as mine.
*I used adapters in that size to adapt EcoTech Marine battery backup to AI Nero 5 (through 12v-to-24v DC boost converter)
I'd just search Amazon for that size female jack and see what you find.
Good luck...
Maybe but does it show the dimensions of the inner pin?
Outer sleeve should be 3.5 mm and inner pin should be 1.35 mm,,, again, assuming your EcoTech battery is the same as the two I have.
In the picture you showed, I don't "see" the center pin so those might use a different method for the negative,,, not sure.
EDIT -- I think these should be the right size,,, but not what I used since I was only buying adapters, not sockets: