It would make sense if the last word in your sentence was a noun (There is no... needs a noun). Nudna is a feminine singular adjective (masculine form - nudny).

If you really want, you might use the noun nuda here (it's feminine; its genitive form is nudy). It means boredom but it's used more often than the English equivalent (and interestingly enough it usually has colloquial meaning). So if you said "W szkole nie ma nudy" it would be correct although, like I said, it sounds rather colloquial while the default translation is neutral. I'd stick to nudny in most cases.

But grammatically it makes no sense, no matter what's the context. The English sentence has to have a subject in the form of "It", but from the point of view of Polish, this is a subjectless sentence. For us, there is no 'it' here.

Well, it's an adverb. It doesn't seem that the English Wiktionary has an entry for it, and it won't be in the declension of the adjective "nudny", as it is not any form of it.

In the English sentence, the subject "It" doesn't really refer to anything specific, right? It's just needed because English absolutely needs some subject. In Polish, this sentence doesn't have any subject. And it uses the adverb "nudno". Similarly "Jest zimno" (It is cold), "Jest ciemno" (It is dark) - they don't have real subjects and they use adverbs.