But that'll just make the phone itself "beep" (monotonically) when the keys are pressed, not affect its (in)ability to transmit DTMF tones, won't it?
It sounds like the OP's wife's phone is just malfunctioning, although I guess there's a chance a different setting got inadvertently changed. My own "Barney Rubble special" (a Samsung SCH350) lets you choose between "normal" or "long" DTMF tones (in the "call settings" menu.) Just on general principle, I set it to "long" immediately after activating it so I don't know if it really makes a difference, but looking for a similar setting is worth a shot...
ETA:
It's not clear to me exactly which menu the setting is under, but check out page 52 of the phone's manual (at least the version of it I found
here), where it refers to a setting called "Send DTMF Tones". If that somehow got switched to "off", that's the problem... If that doesn't work, I think shabby's solution is your only alternative...