You sound very disappointed and obviously very upset about where to even start. Here on Google, I did find that
Capitol Hill Psychiatry is listed as noting that late afternoon and evening appointments are full, but that there are openings 9:00 a.m. -3:00 p.m. so that might be worth a call.
Otherwise, here is a listing of Child and Family Psychiatrists in the DC area that you might just start going the list to contact. It is at:
https://psychiatrists.psychologytoday.com/rms/prof_results.php?city=Washington&spec=5
It is true that many will not be taking private insurance or at least have you pay upfront and then deal with any insurance coverage, but right now getting a definite diagnosis of your daughter sounds paramount. Also, you might still contact National Children's Hospital and/or Kennedy Kriger in Baltimore to see about getting an evaluation done there if appropriate at least to set up a date as a second pathway.
And while she does not know the words, your child may still have the feelings that put her at a danger to herself so do take getting her in to be seen by someone as an immediate priority.