It seems undisputed that the Toyota was attempting to turn right. Indeed, the vehicle photographs show that this was an angled impact in which the Kia's central right front struck the Toyota's right rear corner.
The skid marks appear to be post-impact tire marks. Matching those up to the tire that made them is key to placing the vehicles on the roadway.
If I were a betting man, which I am not, I'd wager that the tire mark is from the Toyota's left rear tire. If it is, then that would place the driver's side of the Toyota near the lane line, which combined with the angle of contact, is consistent with it turning right from the left lane.
With the evidence of a significant impact in the form of vehicle damage and post-impact vehicle motion, I'm wondering what the speed limit is on that roadway and how fast the Kia was going?
I don't know how much resources you are willing to commit to get closure for yourself. An easy thing you can do is to find out if the investigating officers took scene photographs. Obtaining a copy of these photographs would presumably assist in identifying the source of the tire mark.
Depending on the model and the year of the Toyota, it very well may have crash related data stored in its airbag control module. If pre-crash speed/brake/throttle are recorded, it would either support or contradict the Toyota driver's accounts. If the Toyota's driver and/or insurance company does not want to grant you or your representative access to the Toyota, you are not likely to get it. You would have to go through your insurance company.
Similarly, depending on the model year, your Kia Soul likely recorded a crash event, which would report crash severity and pre-crash steering, which may be helpful in piecing this crash together. The Kia will also report pre-crash speed and other parameters, such as if the driver's seat belt was in use, so you do have to think about how important it is to you to know ALL the facts. If you are going to get this done yourself, you have to do it before you sign the vehicle over, or you could harvest the airbag control module before signing the vehicle over. Otherwise, you will have to have your insurance company take care of that too.
The reality is that from your insurance company's perspective, their insured's car struck another car hard causing significant property damage and injuries presumably in both vehicles. The police report description places the blame on their insured. Based upon the information available to them, is it really wrong for them to pay those claims? Wouldn't you rather have the insurance company quickly take care of the people and the damage and sort out fault later.
Unfortunately, these type of situations feel out of your control, because they largely are.
Circling around back to what I think is the original issue, it does appear that the physical evidence validates your daughters account.