It'll probably be a problem on the power supply board on the left there. At a guess its the component to the left of the electrolytic cap that looks like a tall transistor bent over, but is in fact probably some kind of voltage regulator. It might be labelled Q1. Good luck finding a replacement.
It occurs to me that someone could make some money selling replacement Speak power supply boards, as its always something on there that dies.
The amount of resistors doesn't matter. Theres something like 10 or 15 different versions of the speak&spell board, all with different numbers of resistors on the bottom right.
Alternatively, try getting it to start up by just shorting the pins where the keypad connector plugs in. I'm not sure which pins it is but two of them emulate pressing ON on the front panel. I'm sure this has been discussed before.