For me, the most likable character in this story is actually John Clay, aka Spaulding. This is strange seeing as how he is the villain and is described as a murderer (among other things). However, I think he is the most interesting character.
Holmes, as the first answer suggests, is too obvious. Watson and Wilson seem to me to be kind of similar characters. Neither of them is particularly intelligent, it seems. They both seem pretty dull and not very interesting.
Clay is the only remaining character who we see enough of to care about. To me, he is interesting because of how clever he is. When I think how complicated his plan was, I can't help but be taken by him. To come up with the idea of the Red-Headed League is really pretty brilliant. I also like him because of how hard he has worked for Wilson. This endears him to me because he's not being gratuitously harmful to anyone. As long as he's "working" for Wilson, he actually does his job and does it very well.
So Clay is intelligent and in his way a nice person. That makes him interesting to me.
It seems like the nature of this particular question is one that is subjective, driven by the personal opinion of the reader, or in this case, the student. You could gauge where other people on enotes stand on this issue, but in the end, it sounds like you are going to have shoulder the burden of writing it. I think that there can be some leading questions that might be able to help you out in composing the writing based off of the reading. The first question would be the most elemental: Which question struck you as the most likable in the story? Perhaps, this takes on multiple forms such as which character were you able to identify with the most on a personal level or within the parameters of the action in Doyle's work? Who struck you immediately and represented the person for whom you held some vested interest? Maybe the flip side of this question could also help. If you are able to identify the character you dislike the most, perhaps finding their opposing force might allow you to find them to be the most likable if only because they were poised against the individual that you found yourself against. In analyzing this topic, personal reflection combined with reflection about the text will be of vital importance for you in the composition of this writing sample. One last point for you here. When you start writing this, the most obvious answer to the question would be "Sherlock Holmes." Certainly, there is a reason for this. Yet, I would think that you might be able to distinguish yourself in the eyes of your instructor if you chose a character that not everyone else chose.