Bring in or bring into

Whether to use in or into with bring depends on the presence or absence of a prepositional object.

Consider "He brings in the papers" or "He brings the papers in". There is a direct object (papers) but no prepositional object: where he brings them is not specified other than by context.

If you bring the papers into a defined space or area, that destination is a prepositional object and requires into: "He brings the papers into the room"; "He brings the papers into London".

The same principle applies with in/into your life: here, your life is the prepositional object and into is needed.

With a sentence like "He brings the papers in London", there is no destination; in here means within. Someone else would bring papers within Birmingham. While this is similar to your first example sentence, your intended meaning is not the same. You actually want happiness to end up in your life, whereas with my sentences here, the papers are already in London and simply being brought within that area.

[Note that prepositions can vary idiomatically. Bring into works as described here; enter into (for example) works in a similar way, but not identically and has a different meaning.]