I have a Honda Jazz 2003. The keys are not remote but maybe are worn so much that it will not start  the car when inserted in the ignition. The key will open the doors though. How can I overcome the problem, do I have to buy a new ignition  steering lock?

If it will not physically fit and turn the ignition lock then it would more than likely require a replacement barrel.

If you contact your local dealership they should be able to match one up to your original keys by special order or if not then you would have two keys which is not too much of an inconvenience really.

It may be worth spraying some WD40 or similar into the lock and try the key again before committing to ordering parts