Why are Else and Lil so close?
When people suffer from social alienation, as it is in the case of some cultural groups for example, the observable tendency is that those who have been alienated will come together, using their social dejection as a common identifier.
From the evidence that is found in the story "The Doll's House", it is safe to conclude that sisters Else and Lil are not only socially alienated because of being poor, but that they are also neglected at home. Considering the nature of their adult support systems, it is no wonder that the girls have turned on each other to be able to surface through life.
They were the daughters of a spry, hardworking little washerwoman, who went about from house to house by the day. This was awful enough. But where was Mr. Kelvey? Nobody knew for certain. But everybody said he was in prison.
Therefore, it is the combination of social and personal needs what brings the sisters so close together.
Notice the huge difference between the way in which Kezia and her sisters are "protected" from the likes of Else and Lil while the latter, in contrast, seem to roam around free and land wherever they land. Furthermore, it is also evident that Lil has taken a form of motherly role while Else looks up to her for protection. This is evident in the following description of the girls' dynamics
[Else] went through life holding on to Lil, with a piece of Lil's skirt screwed up in her hand. Where Lil went our Else followed. ... Only when she wanted anything, or when she was out of breath, our Else gave Lil a tug... and Lil stopped and turned round. The Kelveys never failed to understand each other.
Considering that Else has obvious social issues that have been left unattended, Lil has taken the initiative to simply respond to the needs of her sister. This is more than a mere social need: it is a survival skill. In different terminology, Else and Lil have had to re-format the structure of the family unit in order to compensate for every basic need that they lack: consistency, protection, and nurturing. Therefore, the girls need to make it somehow. It is in their connection that they find the strength to survive within a society which renders them incapable to make it otherwise.