If it happened to me, I would dig out a length of similar mains cable and plug and splice it on to the remaining end of the Dyson. You may have on unused appliance around that you could nick it from. The safest way to connect is to obtain some in line crimp connectors, the ones that look like little tubes. You might have the crimp tool around or you could borrow one. If you have to buy one your local marketeer / tool stall has them for a pound a kit. For the professional finish try to get some heat-shrink tubing (with adhesive lining - e.g. from Maplins) but slide this on the cable before you connect, use your paint stripper gun to shrink it onto the finished connections. To reduce the bulk of the connectors they can be staggered but it will be stronger if they are clamped together.
Malcolm X, September 2007