I do not see a need for you to sign anything else. Once the property is transferred to you, it will be your property & your partner will not have any interest in it. However, just to be on the safe side it is perhaps best to have an agreement drawn up by a solicitor (maybe the one that you use to do your conveyancing of the property), just to set out the terms that you have agreed, ie. the payment of �10,000 and that your partner will not have any other interest remaining in the property.
I would also suggest that you think about whether you have any current rights over the property. Have you made any contributions to the mortgage? Any home improvements? Although the property is in the sole name of your partner, if you have made any contributions to the household...in the past, the courts have found that the property is held on trust by the sole named partner for both in the relationship. This entitles you to a share under a resulting or constructive trust.
This is perhaps something that you should look into as well.