So, the cunning plan.
I bought a similar voltage/power battery off Ebay, cheap.
Dismantled it.
Used the cells in the original laptop battery.
Job done!
Or not.
Unfortunately these cells refuse to hold a charge longer than 5 minutes either.
Plan D is to somehow try to fabricate a power pack which can hold ordinary rechargeable AA batteries. I’ll post an update after I’ve blown myself up.