How does she find herself in a situation where she has enough Fae connection to assume a Queen Mantle?
1) We don't know if she does/doesn't have fae roots. Molly was several generations distant and needed 'rehab' to make her into vessel.
2) She has been at Arctis Tor, been in NN, fought several Fae, killed them, killed Winter Lady, trains with Einherjar whose Lord has a Mask of a Fae King (Kringle), knows that Immortals can be killed on Halloween, knows that Molly was shaped into a vessel for Lady...
Also worth pointing out that close association with Winter is a big part of preparing an individual for a queen mantle. Per Cold Days:
There was one other person there who had been spending time with a powerful fae.
Who had a relationship with one that was deeper and more significant than a casual or formal acquaintance.
Whose life had been methodically, deliberately, and covertly reshaped for the purpose.
Who had been extensively prepared by one of the Sidhe.
Harry is a part of Winter now. He has already spent a lot of time with Murphy and will presumably be spending a lot more in the near future (some of it pretty intimate). He has been training her and shaping her since they first met. Arguably, all that is left is some deliberate preparation by a Sidhe. That may happen in the past during the events that lead to the vacancy in the Winter Courts.