The battery within that unit is likely no good anymore.
the good news is that they are usually very easy to replace.
These are the two most likely batteries within that unit:
https://www.amazon.com/Sealed-Lead-Acid-Batteries-UB12120/dp/B004JHI8MM
https://www.amazon.com/s/ref=nb_sb_...earch-alias=automotive&field-keywords=Ub12180
You can hook solar up easily.
The plug in wall chargers that come with these jumperpacks are just single stage/ voltage wall warts which can barely, and only with enough time, get the battery within to a fully charged state when it is still newish and healthy. When older and degraded 13.8v applied for days will not really be able to return the battery to maximum energy density as a higher voltage is needed to force the last few % into it.
A solar controller will take much better care of the battery than will the 900Mah 13.8v wall wart provided with the unit.
Those AGM batteries above, when deeply discharged, Ideally should get close to 4 amps initially appled for the 12Ah battery and 6 amps for the 18Ah battery, until voltage gets to 14.5 to 14.9v, then held there until amps taper to a very low level(0.5% of the capacity), then drop to 13.6v float.
13.6/.8volts applied by the provided wall charger is a far cry from ideal for recharging these batteries. but is a OK safe set it and forget it charger when the battery is healthy and the time to recharge is not a factor.
Deep cycling and often cycled AGMS want a different recharge regimen, and that means higher amperage to a higher voltage applied promptly after a discharge. 100 solar watts is about 5 amps at noon.
You need not pursue ideal, these batteries are pretty inexpensive. Ideal just makes them last longer and perform better while they last.