Your electric tongue jack is powered by the Airstream's battery(ies). If they are operational, you should be able to operate the jack using the UP/DOWN switch.
You might also be able to operate the jack using the 12 volts provided by the converter. Just plug the shore power cable into 110 volts. However, this might not work, if the batteries are completely dead; in which case, you may need to buy at least one battery for your Airstream to be able to operate the jack electrically.
If all else fails, electric tongue jacks have a manual handle that can be inserted into the motor housing to hand-crank the jacking mechanism. Hopefully, the previous owner knows where it is. Otherwise, you may be on a quest for the "holy grail" (i.e., a jack handle for an old electric tongue jack).
One last resort: Use a tire/wheel jack to raise the tongue of the trailer. After our GYM blowout and stranding, we carry a two-ton floor jack in our pickup for special problems like this.