I’m reminded of those who “feel badly” about things.
Someone is correctly described as “feeling badly” if he’s trying to feel up his date and is doing a terrible job of it. But if you feel bad or sad due to some misfortune, “feel” is not an action verb here but a description of a state of being; it’s technically a linking verb called a copula or copulative verb that links the descriptor back to the subject, and the adjective form is called for.
Having said that, this is different! “Wrong” is both an adjective and an adverb, and in this case is doing honorable duty as an adverb following an actual action verb. Thus, in the unfortunate dating situation cited above, the accosted female would be correct in saying “You’re doing this very badly”.
Why don’t we say “you’re doing it wrongly”, which is also correct? We could, but common usage has relegated “wrongly” to generally formal contexts, which is why it would sound a little odd to most of us and why we tend to associate it with legal usage, like “wrongly convicted”. Yes, I know, I should have said “such as” instead of “like”. Suck it up. 
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A few more random little things that may be of interest. At this point someone might say – especially in spoken conversation – “Well, did you think that was informative?” or “So, does anyone have anything else to add?”. So, why do we do this? What’s with the "well"s and the "so"s sprinkled like decoration all over our sentences?
I must admit that some people do it so often (especially with “so”) that it’s annoying, but (within reason) it’s neither wrong nor redundant in common speech. Try omitting it and see what you sound like. The words are a lead-in, a softening, an acknowledgement of the other person that creates a less formal and more amicable mood in the conversation. To a linguist, they would be examples of pragmatic markers, words or phrases whose value is modal (literally, “expressing mood”) rather than semantic. They add no meaning, but they enhance interactive discourse.
The final thing I’ll throw in here is a couple of comments about the uses and abuses of “I” versus “me”, the first of which is a rehash from another thread. Most of us cringe at the hypercorrected wrongness of “between you and I”, but what about “I wish everyone was more like I”? Most of us would say “I wish everyone was more like me”, even though the subject case seems the right one because of the implied “am”: “… more like I am”.
Turns out, we’re not the only ones who like the object form here. As Mark Liberman points out, so did Shakespeare in Romeo and Juliet: “Doting like me, and like me banished”. Jonathan Swift, in The Lady’s Dressing Room: “He soon would learn to think like me”. Robert Southey, in his poem To Lycon: “And train the future race to live and act like me.” Charles Dickens, The Old Curiosity Shop: “… he told me about my mother, and how she once looked and spoke just like me.” As Liberman discusses, “like” takes a noun-phrase complement (“handles like a shopping cart with a wobbly wheel”), so as all these writers instinctively knew, the object case is the one consistent with this usage.
Then we have the interesting case of “Jane and me went to the store”, for which the teenage perp would likely get his knuckles rapped by an English teacher. I prefer “Jane and I” in this case, but Steven Pinker raises an interesting point. Forget about “I” vs “me” for a moment and change the tense, and you get “Jane and me are going to the store”. For the sticklers who insist we would never say “Me is going to the store”, one can point out that we would also never say “Me are going to the store”, yet here we are happily using the plural form when everything inside the phrase is singular.
The moral of the story is that the pronouns inside a conjunction phrase should be able to have a different case from the case of the phrase itself, just like the phrase can have a different grammatical number from the pronouns inside it.
All of this can be cheerfully acknowledged while still deploring abominations like the figurative use of “literally”. Don’t get me started – not in this thread, anyway. 