I cannot click "Deliver Work." I see no such button on the screen; only "Give a refund" is an option. I have discussed this with my client, and he is willing to pay, but there is no mechanism for me to receive payment. What do I do?
I checked the job you are referring to and it looks like it was closed, and the amount in escrow was returned to the client's account. For legal reasons Upwork isn't able to keep funds indefinitely. A fixed-price contract with funds in Escrow will be closed automatically after 90 days of inactivity, and the funds will be refunded back to the client's account.
I would suggest that you request the client to issue you a bonus instead for the work you have completed. Or a new contract can be created, a milestone set and funded. And on the contract page, click the "Submit work/Request for payment" button so that you can be paid. You may read up on this help article to know how you can be paid for fixed-price contracts.