I am the only DIL. For many years, I had a close relationship with my MIL, but I don’t know if it was the one she wanted. She encouraged me to call her Mom or by her first name, but I felt uncomfortable with that, so I generally refer to her as Grandma. It feels kind of odd to me that D’s bf of 2 years calls her by her first name and that she loves it! I still call her Grandma, as do my kids.
When my SIL moved back in with her mom after divorcing, it put a real damper on our relationship. My SIL is a toxic person and I can’t compete with a mother’s love for her D. MIL and I used to go out together several times a month, just to Costco or the like, but once SIL came home, it was no longer fun and I just pulled back. I know Grandma is grateful to me for a couple of things I have done and that SIL hates me for the same things, which involve financial moves intended to provide for MIL in her old age and protect her money from her D, who is a substance abuser. I have taken charge of MIL’s finances and oversee her medical care and the like.
I may not call her every day or even week or tell her I love her or hug and kiss her when I see her, but I feel that I do my part to demonstrate how important she is to the life of my family by the tangible things that I do to make sure that she has food, shelter, medical care and 24/7 home health aides. I am not a physically demonstrative person so for people that are, like my MIL, I may be off-putting.
I don’t have kids in law yet, but oldest son has had the same gf for 10 years and D the same bf for almost 3 years. S’ gf is very shy and reminds me of a lazy, unmotivated version of myself. She is not physically demonstrative and that doesn’t bother me in the least, in fact, I prefer it to the way D’s bf, whose FOO is not American, is always hugging everyone, including H and my sons. He has seemingly picked up on (or D had told him) of my preference not to be hugged all the time. Other than that, I adore him. In terms of money, H and I are the wealthy parents compared to S’s gf’s family. Her dad is deceased and her mom is in a nursing home after a devastating stroke in her 40’s. She lives with her grandma, a retired government employee. D’s bf’s family is more like ours in terms of education (3 out of the 4 parents are lawyers and the last is a mental health professional) but I think they are wealthier than us. I spent a couple of weeks with D, bf and his parents this past summer and his mom and I both grew up very poor in our respective countries and have worked hard to do better for our children. We bonded over that. I feel like my D would have a wonderful MIL situation if she married this young man and a far better SIL deal than I have as well.
Bottom line - K may naturally be less demonstrative than M. It’s like the book, Five Languages of Love, which I think applies to every relationship, not just marital. OP may prefer being shown the affection of her children and children in law in her dominant language. It seems M may speak that language and that K does not. I realized after reading the book that my language differs from H’s and 4/5 kids, as well as MIL’s. I try to accommodate to accept their differences and work to adjust myself to theirs as much as possible.