Assuming you are in England or Wales and your daughter isn't at a special school, getting started really is just as simple as submitting the letter.
After that, give her some time to recover from her difficult experiences. Let her play and relax and do whatever makes her happy. You'll only have a couple of months until the end of the school year anyway, and she'll be in a better state to learn in the autumn once her mental health has improved. If that doesn't feel comfortable to you, you could ask your daughter to identify some topic which interests her and let her do a project on that; you can cover most subjects through project work.
If the LA get in touch with you, which is likely, tell them that you'd like all communication to be in writing. You can post back here or on another home ed list if they ask you for anything, and we will be happy to point you toward the relevant laws and suggest options. Some LAs misbehave and try to demand that parents comply with their made-up policies which are not compliant with the law, such as telling you that you must have a home visit.
When you say that you want your daughter to remain on secondary waiting lists, does that mean that she doesn't have a place at your preferred school? Do make sure to tell the LA that you are planning to send her to secondary school in the autumn and want to stay on the waiting lists. They shouldn't assume that deregistering from primary school means that you don't want a secondary place, but it's best to be sure that they don't make that assumption and cause you hassle.
You could make contact with local home education groups to see whether there is anything fun going on which you'd like to join in with. People in my area organise educational tours to wind farms and the crocodile zoo, and there's swimming, bowling and cheap ice skating, for instance.
Good luck with it! I hope your daughter feels better soon.