First, regarding the word 「みたく」, it is dialectal and fairly slangy. The majority of us native speakers do not use it actively. Those who do use it would tend to be from around Kanto and "not very old". If a J-learner wanted to use it, I feel s/he should use it knowing it is indeed that type of word so that it will not be used at the wrong places.
「みたく」 means 「みたいに」 or 「のように」, and not 「みたいな」 or 「のような」.
"Just as Takeshi appears to, I have 'May Sickness.'" ?
Strictly speaking, no, it does not. At least, that is not how the more careful speakers would use 「みたく」.
Unless it was said as a joke, one would need to know for sure (100%) that Takeshi has it to say that Japanese sentence because not many would appreciate being taken for a mental illness patient if they are not. How Takeshi apppears to you should not matter. It is different from a situation where you might be saying "Just like Takeshi, I have a broken arm."
If #2 were true, then does "Takeshi and I are similar" become implied and would this also be a good translation:
"Takeshi and I are similar in that we both have 'May sickness.'"
I myself would not call that a good translation because it is more a paraphrasing than a translation. The subject of the original is definitely the unmentioned first person; Takeshi is no part of it. In that "translation", it is "Takeshi and I". It would be a very free translation.