Use your best judgement as PIC, and be able to explain your rationale late if asked. That said, if you've decided to continue a flight with unserviceable equipment, adjust your technique to make up for it, and do your best. You certainly should have adequate skill to fly a circuit, and land with no airspeed information.
One anti bug trick which has a slight chance of working is the use of pitot heat, which might heat out the bug bits, but don't depend upon it
The GPS could be quite useful for speed information in cruise flight, but I would put it away on final approach. It would be too distracting to be glancing at it, out of position, and it might have an update rate which is too slow to be of use.
If you need speed information, and you're in communication with a radar controller, they can tell you your speed, workload permitting.
You will encounter many odd "little" problems with planes, always be ready.....
