I would be complaining lots too.
I am sorry this has happened to you.
There is no way you should be in for another 11 week wait period, especially since the error was not yours.
It is true that some LAs trynot to recognise private dx, but you need to examine what you want from your dx. LAs get involved at statement level - if what you want/your dd needs is recognition from school, and school are likely to be proactive in their reasonable adjustments, then you don't necessarily need the LA to acknowldge the dx, iyswim?
As to the question of wait times - with dd1 I had interminable waiting lists. followed by a lengthy dragged out dxing process. In all, she was in the system for 2 years before being dx'd (at the age of 2.8) with severe ASD.
With dd2, I opted for the private route, as the dx process on the NHS would have been incredibly lengthy, as is quite a subtle and complex case. Done and dusted in about 10 weeks from first enquiry. She was dx'd earlier this year, at age 7, with AS.
With ds, I have had much shorter waiting list times - nearer the 11 weeks you had - and the system seems to be working well and on time (ish - his latest paed appt is about 7 weeks late). However I expect to be on the rollercoaster for much longer this time around, as he is not presenting as severe atm. He is 2, and has already been in the system for 9 months or so.
As to private costs (in case you want to look into it) - we saw Margo Sharp, who is excellent, and experienced with dxing girls. It cost us abut £1600 for her (whole day) visit and report.
If i were you, and worried about having a dx in time for secondary transition, I'd be tempted to get a private dx while staying on the NHS lists and waiting for that to follow (so that you have the 'acceptable' dx if it comes to that), if at all possible.