What about the boarding house rules made it hard for Lyddie to have Rachel with her in Lyddie?

The boarding house does not allow children to stay there unless they are workers. 

Lyddie works for the factory, and stays at a company boarding house.  The boarding house is for single girls, and there are no children allowed.  Lyddie is alone when she first gets the job, and it never occurred to her that she might need to have a child with her until her uncle showed up with her little sister Rachel. 

When Rachel arrives, she is in bad shape.  She is skinny and ill, and she is not strong enough to work.  Rachel’s mother is dead, and her uncle does not want to keep her so he takes her to Lyddie.  He likely wants to be free of the burden of this family.  Fortunately for Lyddie, Mrs. Bedlow is understanding. 

She could read the warning in Mrs. Bedlow's eyes. No men, no children (except for the keeper's own) in a corporation house. But surely the woman would not have the heart . . . 

"I'm begging a bath for her. She's had a long, rough journey in an ox cart, and she's chilled right through, ey Rachie?" (Ch. 15) 

Lyddie offers to pay full board for Rachel and reminds Mrs. Bedlow that she will not eat as much as a grown woman because she is so small.  She asks for just a few days to sort it out, because Mrs. Bedlow tells her that there is no way the girl can work as a doffer.  She is too small and weak for even little girls’ jobs in the factory. Mrs. Bedlow agrees.

The factory controls many aspects of the girls' lives.  To maintain its reputation, the company makes the girls go to church, dress well, and follow a strict code of ethics.  The company provides housing, at a cost, but does not want its workers to be mothers. 

Rachel decides to appeal to her brother Charlie.  Charlie works at a mill, and lives closer to the family farm.  She tells him about Rachel, and begs him to find a way to keep the farm.  They do not want it to be sold. 

I don't know what to do with Rachel. Children are not allowed in corporation house. If I can I will take her home, but I got to have a home to go to.  It is up to you, Charlie. Please I beg you stop Uncle  Judah. (Ch. 15) 

Charlie responds by showing up unannounced.  He tells Lyddie that he will take Rachel back with him, because the family he is working for have as good as adopted him, treating him like a son and not an employee.  He even goes to school.  Lyddie wants as good for Rachel, and grudgingly lets her go even though she feels like she has lost her last chance to get the family back together and back on the farm.