How to state the negotiation failed? [closed]

It depends on a number of factors, namely the tone / emphasis you wish to convey. If you are suggesting negotiations were made on the recipients behalf, but an agreement could not be reached (without inferring failings on the side of the party representing the recipient), why not say:

"I regret to inform you that "name of company" is currently focussed on identifying those students with a background in CSE/IT. It is with regret that negotiations in this instance unfortunately failed to reach an agreement"

As noted in the comments, the use of the word work suggests that the attempt / process of negotiation failed and not the outcome, as such I would tend to avoid it.

With that in mind, is it even necessary to mention negotiation happened at all?

Unfortunately it is with regret that the company has decided not to pursue your application further at this time, having decided to only consider those students with a background in CSE/IT.

"Didn't" is an informal contraction of "did not". It would be out of place in a formal communication, and out of place compared with the tone of the rest of your statement.

"dint" is a rare slang/dialect contraction of "didn't" and so is going to be a) even more out of place compared with the rest of the statement and b) extremely hard to understand for anyone who is not a native English speaker (and possibly even for those who are). Most people will consider it an error. There is a real word 'dint" but it doesn't mean "did not". "din't" is not a valid English word, and wrong in all contexts.

Replace "dint" with "did not".