@Schlobbob women have two X chromosomes, one from their mother one from their father. Men have one X chromosome which always comes from their mother (and a Y chromosome from their father). So yes you're right that one of the DD's X chromosomes will come largely from the MIL, but they also have another from their maternal side of the family.
Everyone (except in some diseases) also has 22 pairs of non-sex-linked chromosomes, (named imaginatively 1-22) one of which comes from their mother one from their father. However as @JamieVardysHavingAParty describes each complete chromosome isn't passed as a whole from one generation to the next but they are matched in their pairs and bits are swapped between them.
There are then additional complications as to whether genes are expressed or not (some genes you only want one copy to be active, so one effectively gets switched off), dominant (you only need one copy to get a trait) or recessive (you need two copies to get the trait).
So as others have said, for most traits, a child which looks like a relative will have no higher a chance of carrying other traits of that relative than any of their siblings.
Is there a specific illness you're concerned about? If so, do you know if the father of your DD is a carrier?
[ @Schlobbob it's not true that the X chromosome passes mostly unchanged through the generations, in each female generation the two X's pair up like all the other chromosomes and swap pieces. It is the Y chromosome which passes largely unchanged down the male line, as it never has matching Y to pair up with and swap bits with (though there are a few genes which are present on both X and Y, and sections can be swapped between these)]