I have settled the claim. To what extent can I recover the settlement costs based on an indemnity clause?

To what extent can I recover the settlement costs based on an indemnity clause?

Some crucial information is missing in your post. For instance, there are no details about the terms of the indemnity clause and the settlement, nor is there any suggestion on whether or how the clause and the settlement relate to each other.

Your reference to a third party reflects that he is not a party to the settlement. In that case, the indemnity clause and the settlement are likely independent of each other because they involve unrelated parties. This means that the enforcement of one does not preclude enforcement of the other. For it to be otherwise, this matter would have to be established in the clause or the settlement.

That being said, enforcement of the indemnity clause is subject to the rule against double recovery if the settlement already provided for recovery of settlement costs. That is, if you recover from the counterparty the item of settlement costs, you are precluded from pursuing from a third party that same item pursuant to that settlement.