Nope, K1 is specifically for fiancees only. However, they use one of hte same petitions for both the K1 and K3, (I-129F) so it causes some confusion.
She can most definitely stay in the US and adjust status, depending on her intent when she entered the US as a visitor. If she had no intent of gettign married and staying in the US,
And Can Prove It (which is key), then adjusting status is usually not a problem.
However, if she quit her job, sold all her stuff, and moved out of her home before she came to the US, then obviously her intent was to stay here, and won't look good at interview time.
For marriage based info on immigrating to the US check out
http://www.visajourney.com and in your particular case, specifically check out the forum for Adjusting Status.