The kings of Westeros have always been Targaryen, despite the latest Targaryen (the Mad King) being crazy and getting overthrown by Robert Baratheon.
King's blood is thus most likely Targaryen blood, not Baratheon.
The Targaryens are also the keepers of the dragons that breathe fire. It's only logical that the Lord of Light and the dragons are aligned.
Since the removal of the Targaryens and the coming of the White Walkers (the latter probably mobilizing massively after centuries of hiding because they now sense that the throne of Fire/Light/Targaryens is weak), the Lord of Light (LoL) probably wants to bring the Targaryens back into power.
But first the Baratheons need to be removed since they feel they have the true claim to the throne. So the LoL uses Stannis as a pawn to get the Baratheons to die out in a dose of epic irony.
- Robert Baratheon dies.
- Stannis Baratheon kills his own brother, Renly Baratheon, at the urging of Melissandra.
- Stannis Baratheon kills his own daughter, at the urging of Melissandra.
- Stannis Baratheon "kills himself" by sending his woefully-outnumbered and unprepared army into a suicide mission. Stannis is a very experienced war commander but the fact that he still sent his men into such an obviously-hopeless situation shows just how badly he got played by the LoL - either in belief of the "prophecy" (which was fabricated by the LoL) or inflation of his own ego. The most delicious irony is that he doesn't die by the hand of a Bolton (even though two tried and failed), but by that of Brienne, who kills him to avenge the death of... drumroll... Renly, his own brother which he had killed.
- Melissandra senses king's blood in John Snow, but he dies and all his blood is spilled, so he might have been a Baratheon, possibly one of Robert Baratheon's bastard sons who he entrusted to Eddard Stark's care.
- Gendry is the only other known person with Baratheon blood still left alive at this point, and he has joined the Brotherhood without Banners that worship the LoL.
So now that Baratheon blood - and more importantly - Baratheon forces, are all wiped out, there is a clearing for a Targaryen with fire-breathing dragons to come back in and reclaim the throne. The LoL played Stannis and used the zealot Melissandra to do so.
Gendry is still out there but he's a bastard, has no forces under him, and has a tenuous claim to the throne. Despite this, the LoL still keeps him close since he's now joined with a group that worships the LoL ("keep your enemies closer...")
And the Lord of Light just happens to be "LoL".