It's not so much that Jaggers likes Drummle but more that he has a sneaking admiration for him. In his capacity as a lawyer, Jaggers moves in a world populated by criminals and so he comes across numerous unsavory—but endlessly fascinating—characters like Drummle on a daily basis. Jaggers knows full well that Drummle is up to no good with regards to Estella, but he chooses not to get involved. Here we can observe Jaggers' detached view of the world, a disinterested, amoral perspective which keeps him separated from the common run of humanity. This is what makes Jaggers such an effective attorney.
Drummle is a truly fascinating specimen of nastiness and provides Jaggers with the same degree of fascination as so many of his criminal clients over the years. That Drummle hails from a well-to-do, socially respectable background makes this all the more ironic. As with Abel Magwitch, Dickens is once more challenging the notion that you can always judge a book by its cover.
Jaggers sees in Bentley Drummle a younger version of himself. He sees the same shrewd and ruthless qualities he believes are needed to get ahead in the world. Jaggers nicknames Drummle the Spider because he seems to be able to get anything he wants.