This will (I hope) leave me with a TV which I can switch between
headphone and speaker modes.

The headphone socket will incorporate a switch. There are 4 terminals on it.

You break the speaker cable at a convenient point and reconnect them,
passing through the appropriate terminals (you may need to install
additional cable if there is insufficient slack to position the socket where
you want it. The terminals are -ve and +ve feed from the amplifier, and the
switched -ve and +ve outputs to the speaker. The internal switch disconnects
the speaker automatically when a plug is inserted.

Christian.