Confused about 'What she likes I like that.'

Solution 1:

This sentence is not standard. It becomes standard if you suppress "that".

• What she likes I like.

If you try to restore the normal order, there is no place for "that". This is an indication that the construction is most probably not correct.

This is a case of fronting the object (emphatic sentences); so, there is no difference with the first sentence as the parsing is again that which you gave, except for the fronting of the object clause.

In spoken and written English we sometimes want to make a strong contrast with something in a previous statement. We can do this with objects and complements by 'fronting' them (moving them to the front of the clause), which makes theme more emphatic.

Examples from Practical English Usage, Michael Swan, p. 217

• What I'm going to do next I just don't know.
• How she got the gun through the customs we never found out.

The second sentence, slightly modified can be correct (mere addition of a comma), as for instance in this dialogue. However, the parsing is different, there is no fronting, the first clause is not the object of "like".

• — What sort of house decoration do you personally like? I am not asking about your girl-friend…
  — What she likes, I like that.