It is not possible just looking at the one photo you have posted, to determine how your treatment is going. My advise to you is that if you have questions regarding the treatment in progress, get an opinion from an orthodontist (or two), and then understand the options going forward. "Damon" braces is a highly well marketed tool, but at the end of the day, it is making the correct diagnosis and treatment plan that is the most important first step, regardless of which method is used to achieve the results. Of course, you also need to have a provider who follows through with good technique, but techniques can vary, provider skill is what makes a technique work or fail.
What you are referring to is "fixed" retainers. They are not necessarily permanent as they can break, come loose, etc. There are a lot of factors to consider when choosing to get fixed retainers, including how the initial alignment of the teeth was, how is oral hygiene, condition of gums and bone, etc. But assuming that you have discussed all this with your orthodontist, generally the lower front teeth are good candidates for the fixed retainer. Upper fixed retainers may not be an option in some people due to the way the lower teeth hit the inside of the upper front teeth. If you do get a fixed retainer, it will still help to have the clear retainers made over the fixed retainer/s.
Invisalign is very dependent on patient compliance. The aligners have to be worn 20-22 hrs/day. It is difficult to tell what any particular person "quits",but you should not start Invisalign if you are not ready for the commitment for the entire treatment time. One of the possible reasons is that some people just find out that they cannot keep up with the aligner wear, and if not worn well they will not work effectively and will stop fitting.
It is not possible to answer your question without examining your bite, evaluating the cause of the gap, and the clearance available to close the gap without moving other teeth. You should consult an orthodontist, but likely you will need either braces or Invisalign, less likely (but possible sometimes) to use a removable retainer-type device. No matter how you close the space, you will need to use a retainer to maintain the new positions of your teeth, because if not, the space will very likely reopen.
You should call your orthodontist and let them know that your retainer is broken. There is no easy fix for this, and teeth may or may not shift in a month, really depends on a lot of factors. But your orthodontist can if possible move up the appointment, that would be the best solution.
