Only sleeping on me

Theres hundreds of members on here who would love a lap cat, so your very lucky. I have 3 cats who have igloo beds each and a huge soft dog bed that all three sometimes lie in, but they all prefer my lap.

Sophie sits waiting for me to get settled because the minute l do she's up and snuggled in. Sometimes all three of them jump up so imagine that?!

This kitten is finding security in your warmth a body sounds.  Yes, it will get better, but keep in mind that she likely had mama and siblings or other cats until you brought her home.  When she is on your lap, she is relaxed enough to sleep.  If not, she has to stay awake to make sure nothing catches her and eats her.

Do you really want it to get "bettter".  Since you just brought the kitty home, she's probably a little insecure and scared.  She obviously sees you as her "security blanket".  I would try to bear with her for a few days or more, and as she acclimates to her new environment she should become more independent.   I have four cats.  My oldest, now 4 years old,  has always been a little clingy.  Sometimes I'd really rather be left alone.  But I don't dare push him away.  I feel that loving him can only make more secure - not the other way around. 

My little Max is 7 months today, but we got him around 4 months and he was very much a lap kitty. In fact, the first few nights I had him I let him sleep in my lap. Then, he would want to be in my lap all day when I was working (from home). Eventually, he branched out as he became more comfortable and now he sleeps in a variety of locations, but still prefers to be on us at night, either between our legs or on my girlfriend's head. 

Basically, it will improve as your cat becomes more comfortable with the environment. Also, be glad your kitty is a lap kitty since some people can barely even snuggle with their cats, sadly (just based on the cat's personality and whether it likes to snuggle). 

Good luck!

Do I dare a bad joke?

Good nobody of you is a Scottish.  Otherwise you would surely immedialy try to catch on you some rheumatism.  To have a extra good use of this your lap cat.     :)

Im no Scot, but my aching right knee (rheumatism?)  did went better when our oldest resident took as his new custom to rest between my knees...