A mortise lock has a cylinder that screws into the lock body. This cylinder can be removed and rekeyed or replaced so your lock will work with a different key. (Click above images to see larger version)
What you can do - Remove the cylinder and take it to a lock shop for a replacement or re-pinning.
First you must loosen the set screw. Look at the edge of the door. You may have to remove a face plate to see the set screw(s). There should be either 1 or 2 screws just above the bolt, in line with the center of the cylinder. If you have 2, the screw closest to the outside of the door is the one holding the cylinder.
If you have the single screw type, the screw may turn clockwise or counter-clockwise. As you turn, the screw will move out toward you. CAUTION: Do not force or try to remove the screw - This will ruin your lock!
If you have the two screw type, loosen the screw about 4 turns. CAUTION: Do not remove the screw. It is hard to get back in and may fall inside your lock body.
When your set screw has been loosened, grip the cylinder and try to unscrew it from the lock. Or you can put the key 1/2 way in to use as a handle to turn the cylinder. CAUTION: Do not force the cylinder to turn. You may break the lock case and ruin your lock!
If you have been successful, the cylinder is now out of the lock. Take it to a lock shop to either get an exact replacement or have the internal pins replaced so the lock will work with a different key.
Then with the door open, reinstall. Carefully line up the cylinder so it threads in EASILY. Do not cross thread. When the cylinder is in and straight, tighten the set screw until snug, not too tight. Check the operation of the lock several times. If all works well, close the door and again check the operation of the lock and your new key.
What a locksmith can do - Many of these locks are around 100 years old and can give you some surprises. You may have a lock that is already broken. It may have the wrong parts. The cylinder may be "frozen" in place from years of not being removed.
An experienced locksmith will be able to service your lock or offer options if a lock is not serviceable for any reason. Replacement mortise locks are still available. They are a close fit, but some work is needed to try to cover all the holes and scarring of the current lock. Marks, Emtek and Baldwin companies make mortise locks.