i was rather hoping someone else would step in and comment, but it seems not.
Ni-Mh cells are charged with a constant current. end-of-charge is sensed by a small drop in cell terminal voltage. once fully charged they can be kept 'topped up' with a small trickle charge of less than 1/20th C.
Li-Ion cells are charged with a current-limited constant voltage. during the initial stage of charging the cell voltage will rise from 3.0 volts (discharged) to 4.2 volts (partially charged); the current has to be limited by the charging circuit to something between 1C and 1/4 C. once the cell voltage has reached the 4.2 volt limit the remainder of charging occurs with charging current tapering off and is usually terminated when it drops down to 1/10th of the initial current limit.
if a Li-Ion cell has a voltage significantly less than 3.0 volts it is usually unrecoverable. the charging circuitry should detect this and stop - although occasionally a cell can be brought up to 3.0 volts with a small trickle charge, after which the cell may sometimes be partially functional (for example, having a reduced capacity).
both schemes may have temperature sensors on the cells, along with safety timeouts.
while both schemes are relatively simple, they are very different. trying to charge Li-Ion cells with the Ni-Mh charging scheme will result in a bad outcome. trying to charge a Ni-Mh cell with the Li-Ion scheme will likely result in the charging circuit detecting a fault and shutting down.
you are unlikely to have much long-term success adapting a Ni-Mh charging circuit to work with Li-Ion cells. you will be able to get some charge into the Li-Ion cells, but not in a way that is reliable and safe. you definitely will not be able to safely leave things charging overnight while unattended.
as i suggested before, you would be better off putting Ni-Mh cells back into the Fluke 123 and then creating an external Li-Ion pack that has its own charging circuitry that operates independently from the Fluke 123.
cheers,
rob :-)