A pair of new Tiagra 9 speed shifters alone are about £175 (obviously varies a bit), a basic cassette would start at about £25 at a guess.
You'll also need a hub that'll take the 8,9 or 10 speed cassette, which would mean either getting a new wheel made or rebuilding your existing wheel. The bit I'm not sure about is whether the the frame would take the hub- hopefully someone else will know. I think the new hubs are wider, but I'm not certain.
It would cost a fair bit anyway.
Another option is to buy some road brake levers, just levers- no integrated shifter, and change gear using some bar-end shifters. Might be an option (I might even have some suitable levers in my bits box- but I don't have any bar-end shifters).
Incidently- are both the shifters Donald Ducked? or is it just the front or rear? Seems unlikely they would both fail.