An easy way to do long, accurate time periods, with short on-times. Electro-mechanical wall plug timers have a minimum of 15 minutes on or off periods. Digital timers have a minimum 1 minute. Adding a DC wall wart with a simple relay-monostable will do the trick. And the output is isolated from the line and DC supply. The on-time depends on the size of the electrolytic capacitor, the coil resistance of the relay, and it's pull-in and dropout voltages.
Actually you could do any type of monostable off the wall wart.
Ken
Oops!. A reread of the post looks like you want it built into the Rumba. Do it with an 8-pin PIC or equivalent microcontroller.