1) A common 18650 Li-Ion cell is stored and sold at the half-charge voltage of 3.7V. It is fully charged at 4.2V and its loaded voltage slowly drops to 2.5V when the load MUST be disconnected to avoid damaging the cell. My products warn me at 3.2V and disconnect at 3.0V.
2) A white 8mm LED in Google has a max continuous current of 60mA and an average forward voltage of 3.2V. Then 2 in series with a 15 ohms resistor and powered from 6V barely light up. One 3.2V LED with a 27 ohm resistor and powered from 4.2V draws 37mA but when the voltage drops to 3V the LED barely lights up.

 

Well......looks like solar might not meet my expectation then
It seems I should consider running wires, with current LEDs or new LED LED lamps such as these ones:
https://www.amazon.com/Non-dimmable...cd3-8ffe-a34859bd0e88&pd_rd_i=B01C8G7PUG&th=1

 

Another option is wires and an external supply of between 3.7 and 4 volts. Probably a 5 volt charger and a resistor for each one and no modifying. You will need to determine the current drawn by each to calculate the resistor value, or find it experimentally. by trying different resistor values. For white LEDs at 25 milliamps each, and pick 3 volts that means about 2 volts drop on the resistor.. So try 47 ohms first.

 

Most of these use a QX5252F controller with an external inductor to kick up the voltage to drive the LED(s). The controller is dirt-cheap on Amazon as are inductor assortments - the inductor determines the power delivered to the LED. The controller manages battery charging and ON/OFF functions. I've fixed a bunch of cheap solar stuff using these and a BOJACK 1/2W Inductor assortment also from Amazon. The circuit is simple enough to hand wire but you should be able to clean the PCB enough to get it working again.

The basic circuit can also be used to add lighting to toys etc. that use a single 1.5V alkaline - think race car with pink headlights for the granddaughter.

Have fun!

 

Given that the batteries have all aged it seems that those will be the big deal replacement item. Thus my suggestion to wire all of the lites to an external supply. Then no battery issues any more.