Quite simply, it wasn't necessary, and would only have complicated and escalated matters. Lawrence Freedman has a good summary of the British government's thinking in the second volume of his official history of the conflict.

As soon as news of the occupation of the Falklands arrived in London, foreign secretary Lord Carrington asked the Foreign & Commonwealth Office's legal adviser, Sir Ian Sinclair, for advice on whether a declaration of war was warranted and necessary. Per Freedman:

According to [Sinclair], this was the most 'momentous' piece of advice he was ever asked to give but also the 'easiest'. Any action Britain now took would depend on the 'inherent right of individual or collective self-defence' as confirmed in Article 51 of the UN Charter.

The primary reason that declarations of war have become something of a legal anachronism is that a formal state of war has significant legal (and economic) implications. Civilians can be detained and property seized; relations with third-party states become complicated, because neutrality as a legal status also brings its own baggage. In Freedman's words: "The absence of [a state of war] might suit both parties to a conflict as a means of keeping it limited and avoiding difficult questions in relation to third parties."

You might find this interesting reading: a minute, titled 'Formal Consequences of a State of War Between the UK and Argentina', that Sinclair submitted to the Cabinet on 9 April 1982. His opening is rather pithy:

Question 1: Do we need to respond by formally declaring war on Argentina?

No.

He goes on to explain the formal process and implications of declaring war in some detail, as well as summarising the underlying law. But his conclusion is clear and unequivocal: Article 51 "clearly comprehends any military or naval action which might be necessary to repel or expel any invading force. As the right is an inherent one, it need not be accompanied by a formal declaration of war."