Mrs. Markleham is also known as "the Old Soldier," because she is constantly maneuvering around her son-in-law. As David puts it, "our boys used to call her the Old Soldier, on account of her generalship, and the skill with which she marshalled great forces of relations against the Doctor" (16.122). She seems always to be waging war against Doctor Strong as she tries her best to manipulate him into finding jobs or money for her various relatives (including, most obviously, Mr. Jack Maldon). As soon as her daughter marries Doctor Strong, Mrs. Markleham turns him into a meal ticket. And Doctor Strong is so nice that he never seems to object to this exploitation.
Where all of this manipulation really starts to get nasty is that Mrs. Markleham (either intentionally or without knowing it) plays on Doctor Strong's growing insecurities in his marriage to get money for her own amusements. After a brutal visit from Uriah Heep, who tells Doctor Strong that everyone suspects Annie is having an affair, Doctor Strong becomes convinced that he is too old to hold the love of a girl like Annie. And Mrs. Markleham (again, whether she knows it or not) confirms his fears.
Mrs. Markleham compliments him because he doesn't "expect, as many elderly people do expect, old heads on young shoulders" (45.11). This is definitely a backhanded compliment: what she means is that she's amazed that someone as elderly as Doctor Strong doesn't try to keep young Annie at home. She is glad that Doctor Strong is constantly sending Annie out in the world to amuse herself (with Mrs. Markleham along, of course, to chaperone her). Mrs. Markleham loves her pleasures, and enjoys profiting from Doctor Strong's guilty insistence that Annie go out and enjoy herself all the time, accompanied by her mother.
Once it is all revealed that Annie has been feeling guilty about her mother's manipulations, and that actually Annie would like nothing better than to stay at home with Doctor Strong, Mrs. Markleham's role in their household diminishes. She still lives with the Strongs, sure, but her emotional manipulations drop to nothing.