You can take the form with you on a visit to DLRP anytime in the month or so before your AP runs out, and extend it in person for free at the AP office in the main park.
Or, you can post your AP, completed sponsorship form, and a letter explaining you want the free 12 months extension. Again, this gets posted to the AP office at DLRP (the full address should be in the small print on the sponsor form) and again, you need to post it no earlier than 30 days before your AP runs out. I've extended my AP by post, I sent it signed for service, got it back extended as requested in less than 2 weeks, and they sent it back signed for service as well, at no extra charge to me .