Sally - how to transfer the whole legal ownership from a sole to joint names is covered in a number of forum threads and our online guidance also explains what's needed -
Registering land or property with HM Land Registry: Change the registered owner name - GOV.UK
You will need HMLR forms AP1, TR1 and ID1/ID3 as appropriate. Panel 4 of form AP1 can refer to "Transfer 1 to 2 - Full market value £250K'
The fee is a Scale 2 fee on half the current market value -
HM Land Registry: Registration Services fees - GOV.UK