Even Li-Ion has been around long enough that people could have dead, older batteries.
If it's Li-Ion, you need to be certain that you do not overly charge them with the trickle charger since it doesn't know when to terminate charge for these. Overcharge and they'll be ruined, possibly even dangerous, but even with NiCd or NiMH, too long a trickle can cook them so the goal would be to just put enough charge in to get your normal charger to work again.
If they're Li-Ion, I'd err on the side of caution and charge them in/on a fireproof area, or better still, open the pack and check each cell voltage before proceeding. If any cell is below 2.4V, abort attempts to charge them. They are no longer safe.
It would help to know the rate of the trickle charger and the voltage of the battery pack. If it's an 18V pack and only 12V trickle, depends on how the charger is designed, if it will float the voltage high enough to put enough charge into the pack to make the regular charger happy. Again it depends on the design and rate of the charger but using a 12V trickle charger for an 18V battery could take many hours to get much voltage increase - but don't assume, measure voltage instead because a modern design that's current regulated, could boost voltage enough and not even take an hour to get them high enough.
I'd be monitoring the current and voltage and stop to see if it looks like you have a voltage consistent with all cells taking charge, or are suspiciously ramping towards a voltage that looks 1.2V too low (or multiples of 1.2V) like a cell is shorted. If it's a Li-Ion pack and it looks like 3.6V too low, abort further attempts until you take the pack apart to see what each cell voltage is.
If a cell is shorted you may need to open the pack and zap those particular cells individually to burn through dendrites. This is only for NiCd or NiMH, never Li-Ion. Zap just means a fair amount of current, at least a few hundred mA but better a few amps, at a few volts (typically at least 6V or more works best). This is realized current not rated. In other words a humble 12V wall wart rated for 200mA, could possibly work because it (usually) has a capacitor after the transformer that provides enough instantaneous current. You only want to zap for a couple seconds, remove contact, then zap again until dendrites are burnt through, "if" you have any to burn as evidenced by a cell showing zero volts.
Keep in mind that these are probably only temporary measures and the pack will have reduced capacity and be a problem in the future. If it's a nice drill and you have time to burn, you could rebuild the packs (if you have a decent soldering iron but technically you're supposed to use a spot welder too, to join the cells instead of soldering to their casings, although some cells are sold meant for battery pack building and have the nickel plated tabs already spot welded on and you just join those with a soldering iron), but otherwise new batteries or a new drill is the easier long term solution. There are good advantages to Li-Ion if they were NiCd or NiMH. Once you switch it is hard to look back.