This quest cannot be completed until you visit Arl Howe's estate to rescue Queen Anora. Once there, he is located in Arl of Denerim's Estate - Dungeon. You get the key to his cage from the Jailer. This can be done before or after facing Arl Howe.
Should you decide to put Rexel out of his misery, the quest will end there. You will not have the option of reporting back to the Chanters Board and will thus receive no reward. In this case the completed quest note will state, "You killed Rexel in Arl Howe's dungeon. His family and Chanter Rosamund are unlikely to discover his fate."
If Rexel is not killed, you may turn it in at any chanter near a Chanter's Board for a 2 reward and 175 experience.