You can add payments to invoices by navigating to Invoices, opening your invoice and choosing "Manage Payments" from the upper right Actions menu. You will then be redirected to a page where you can view, add, edit and remove payments for your invoice, along with descriptions, dates, and more.
On this page you will also see the date each payment was made. If your invoice has been paid in full rather than partially, you can consider the date next to your invoice's sole payment the date your invoice was paid. Each payment comes with a separate payment receipt. At the far right of each entry you'll see the receipt number for the respective payment - click on it in order to open your payment receipt.