What your dictionary is referring to (or should be referring to) as “informal” is not this use but the use of (a) sort of (a) or (a) kind of (a) as adverbs, meaning approximately “somewhat” or “rather”:
That's sort of a mean thing to say.
She's a kind of private person.
He sorta kinda twisted it like this and it popped right off!
Your use, where sort (or kind) is used as a frank noun, is acceptable in all registers.
But you do have to be careful about what sort or kind refers to. In your example, for instance, it’s not clear which of these you mean:
- all print∗ media have a long production cycle, or
- newspapers and magazines, unlike other print media, have long production cycles.
Your use of the plural suggests the latter, but I suspect you mean the former. If that is the case you should recast so that sort (or kind) is in the singular, which in this context can only refer to print:
This sort of medium usually has a long production cycle.
Media of this sort usually have a long production cycle.
Better still is to be completely unambiguous:
Print media usually have a long production cycle.
∗ Note that the medium of publication is print; only the product is properly referred to as printed: a “printed” book, as opposed to a handwritten book or ebook.
Do the needful is Indian English, which has been covered on ELU.
If you're only interacting with other speakers of Indian English then feel free to use it, but avoid it in any other contexts (most Americans and Brits will think it's quaint/uneducated).
In general, the "standard" form is do what[ever] is necessary, but in OP's specific context most likely nothing like that would be used anyway. If you've just asked for an email address, it goes without saying that you want the other person to do whatever is necessary to give you that information.
I may be wrong, but I have the impression that for many Indian English speakers, "Please do the needful" carries a subtext of "This problem is too complex for me to understand or resolve myself, but I have complete faith that you will be able to deal with it, because you are very skilled in such matters"
As I said, Brits and Americans wouldn't normally use any equivalent for such a trivial problem as finding someone's email address. But if the request was for something more challenging (and crucially, if it was from a manager to a more junior worker), "Do what[ever] [you think] is necessary [to solve this problem]" might be perfectly normal. The implication there is that the manager is authorising the junior to do things he might otherwise not be "permitted" to do (in effect, the junior is being temporarily "promoted" for the duration of the problem-solving).
In that context, it should be clear that (to Americans or Brits, at least) any such phrase would probably be considered offensive/cheeky if addressed to an equal in the workplace (if the person asking isn't senior enough to confer temporary authority on you, they shouldn't be speaking to you that way).
Best Answer
This is more about culture than about fluency. Americans prefer direct requests.
Let's work with your example. A polite desi will imply a course of action and request to "kindly do the needful."
This is well received in India, but this tends to frustrate and even confuse Americans. Most would prefer a direct request.
Hearing this, an American will feel they are being treated with respect.