Paul has the orthodox good way, the barbaric
a la basqueversion I did when I was younger, was to do push ups against the wall.. I know, sounds crazy-it is. Obviously not flathanded, but with the hand shaped, fingertips in the wall forming the arch...Regulating the distance to the wall, you moderate the degree of intensity.. however, I'd obviously would take this with a pinch, or two of salt! Done incorrectly could be far more harmful than benefitial, so I'm really telling this more as an anecdote, than an advice. Though.. I actually do have a very strong muscly hand, however I can't really tell how much the push ups did, or just time at the piano
It takes time, but with slow careful practice, the hand should get toned by the correct weight and use of the muscles, but if the case is as sever as you say, then the off keybard methods Paul gave you will speed the process!