Brake cable should only cost a couple of quid, same for the outer (usually sold by the metre). You may need a couple of ferrules (the end caps) as well, which are less than £1 each and a cable end cap (which may be included with the cable).
It's a straightforward job that you can do yourself.
I would replace the bar tape at the same time, i find it difficult to get the old tape to sit right if reusing it (but maybe i'm just crap at wrapping).
-Strip the old tape off and clean glue residue off.
-Have a good look at the shifter and work out where the end of the cable sits and how it's routed. If it looks complicated, take some pictures to remind yourself. (It will most likely not be complicated at all though).
-Remove the old inner and outer cables.
-Measure the new outer against the old one and cut to size (dont cut the inner yet though).
-Put the new cables in place (with ferrules if required) and roughly tape in place against the bar with electrical tape. (There should be a channel in the front of your bars that the cable sits in.
-Attach to the brake and make sure it's working before you redo the bar tape incase you need to change anything.
-Trim off the excess inner (not too close to the bolt, leave about 5cm) and finish it off with an end cap, or solder it if you want to look like a pro

-Stick on the new bar tape. Job done.