Edgbaston girls, KES, KEHS other indies do have small 13+ intakes.
Only KES has a formal 13+ intake and I think even that is very dependent on places arising by chance ("The vast majority of boys enter the school at 11+, although there are a few places - on average between 5 and 10 - for entry at 13+."). We've touched on it briefly in the past, one year there was just one place and the year after that they had 5 places. I imagine Bromsgrove would have a 13+ intake, otherwise the only school in the area that I know of with a formal 13+ entry is Old Swinford Hospital.
Many schools have 13+ formal intake, the main intake at Bromsgrove is 13 as their prep school is attended until that age. In addition the following have formal intake , RGS, Kings ,Malvern College , MSJ and OSH but to mention a few