If you have any electronics experience, you could try splicing a new plug (or the existing plug if you can remove the failed area) onto the adapter with a soldering iron, but if you don't have electronics experience or you keep using the brick as it is, you run the risk of shorting out the power brick and potentially causing it to overheat, burn out the power circuitry, short out your electrical outlet, and in rare cases cause the brick to catch on fire. If you can't fix it, get a new brick and stop using the old one.
If you do have to use the old brick to keep your laptop going until an appropriate replacement can be obtained, only use it in a stationary environment (set your laptop on a desk and don't move it while it is plugged in, you don't want to knock wires together and create a short) and pull the AC plug as soon as you notice any problem (watch the battery icon, if it stops charging when it shouldn't or says it was unplugged, pull the AC from the brick to avoid a short).