There are two alternative routes forward, both of which come to the same outcome. You can either put your child's named down on the Secondary Common Application Form (SCAF) for a grammar school; they will not be offered a place and you can then appeal, when arrangements will be made for the child to take the test, probably in May.
Alternatively, you can make a late application for a grammar school directly to the school on 22nd April, which is technically called an In Year Application. In Kent you apply directly to as many schools as you wish. Arrangements will then be made for your child to be tested. Some parents will also use this process to apply to further grammar schools, other than those on the original SCAF.
In Kent, which ever of the above routes you have chosen, arrangements for testing are down to the individual school. Whilst the tests will be of similar format to those taken in September, the child's older age will be taken into account so that the pass level is of the same standard.
If successful, and the school of your choice still has vacancies (likely to be around half a dozen grammar schools , mainly in East Kent) you will be offered a place. However, in most cases, the schools will be technically full and you will need to go to appeal. This is likely to be after the main appeals, so there will be pressure on places, but you still have chances of success, depending on the school. If unsuccessful, you still have the right to appeal. Again, I can advise on this.
Medway Council is more difficult to deal with and I find that procedures differ according to the grammar school concerned and the source at the Council providing advice. You should contact the Council and the school, but don't necessarily be put off by discouraging information.
Each year I am approached by parents whose children have underperformed because of illness and the only remedy is to hope that an appeal panel will take this into consideration when considering an appeal in May or June. Some schools will tell parents their children are obliged to take the test in any case. In such circumstances contact Kent County Council Admissions Department immediately for advice.