The jack will not work without a battery. I played that game with my jack for about a year, thinking it was the battery. Charging it, testing it, charging it again. In the end, it was the jack that was slowly freezing up (rust). Finally, I could not even turn it manually.
It is true that the light will work on minimum power from the battery and the jack will not. Hopefully it is just the battery for you.
2017.. Prime Time Avenger. 32QBI
His.. 2011 F-350, King Ranch, 6.2L. Crew Cab, 4x4, 8' bed, SRW, 11,000 GVWR
Hers..2008 Ford Expedition, King Ranch Heavy Duty Tow, 3.73 gears
Sold.. 2005 Star craft, Travel Star 18SB
Sold.. 2013 Keystone Laredo, 291