Really! They say they increased the cost 2 months after your grandmother moved in, never received the money for the next 22 months and never queried it until now? Ridiculous! Who would run their business like that?
Was the cost increased at all while your grandmother was in the home? If so, how was this done?
When the Council became involved was the cost in the contract with them the same as was being paid under the self-funding? If so, you need to get evidence of this from the Council & provide it to the Court as part of your defence.
The claim of a phone call to you is absurd - it is what is in the contract that matters. In any case, if they had phoned you and you agreed, why did they not put it in writing & start invoicing you for the money?
You must defend the Court claim (but make sure you do it within the time limits) & produce all your evidence in writing as part of your defence. The Court should then send it to the claimant who - if they have any sense - will then withdraw their claim. Otherwise make sure you attend the hearing with all the relevant papers etc.