To splice the old broken piece, you will probably need two PL-259 Coax connectors.
Not rocket science to solder these guys (see How to solder a PL-259 Coax Connector | The Steven Experience
Once you have done that you will probably need a Barrel Antenna Coax Connector to join the pieces.
And, you will then have an old piece of coax repaired.
To replace it you will probably need two PL-259 Coax connectors (see above) and some new bulk coax. Try some place like HRO Ham Radio Outlet - World's Largest Supplier of Amateur Radio (Ham Radio) Equipment. Sales, Supplies, and Service.
. Once done, you will have a new piece of Coax. Same job, you will have learned how to install coax connectors and your job will be a better one.