Do you happen to know whether they have any vacancies? (That could affect the sort of case you need to present.)
QEGS is not full in that year group so this will be a non-qualification appeal only
If I were you, I'd get the appeal form in straight away. You can write on it "evidence and details of case to follow" if you need more time.
The point is - for an in-year application they have a maximum of 30 school days from the date they receive your appeal to arrange a hearing. Depending on when their holidays are, this takes us from today up to approx. 3rd May. Admittedly, they then have up to a week to give you the decision, but if you were to explain that your daughter is out of school, it's quite possible they will do all they can to get the decision to you quickly.
I think you probably need to take out the period between 27th March and 15th April for Easter holidays (only school days count for the 30 days) so you are probably looking at mid May. Decisions are sometimes given very quickly (next day or even sometimes same day) in Lincs though I am not sure about this particular school which I think still arranges its own appeals.
I would advise writing something very brief on the "Reasons for Appeal" part of the form, such as "We believe that x has the required ability for this Grammar school despite the fact that she did not reach the qualifying standard in the tests. Full details and evidence to follow." This is simply because I have heard of a form being returned to an appellant who had written "details to follow" as the School Admission Appeals Code says reasons for appeal must be given in writing (it was not an appeal for this school).
It would be be dishonest to say that this will be an easy appeal as 205 is a long way outside the range where non-qualification appeals generally have a reasonable chance of success. However, you have the advantage in Y7 that you should be able to access some evidence of ability which is directly comparable to the ability of the children already in the Grammar school. You should be able to get up-to-date curriculum levels from the current school and should ask the admission authority at QEGS to supply you with the average levels of their current Y7. Secondary schools almost invariably administer some form of ability test (often CATS or MidYIS) to their new entrants but these are purely to help the school set progress targets etc so the results are not reported to parents. You should be able to ask your current school for the results of the tests and ask QEGS which tests they administer and what the average attainment of the current Y7 was in the tests (there is a battery of them).
There are some Grammar Schools in Lincs which would automatically supply information about the attainment of their own school's cohort to an appeal, but this is by no means universal so you would be advised to request it well in advance of the appeal.