As a good match becuz house Lannister was very powerful and Jon arryn wanted to consolidate power around Robert

Tywin by sacking KL, killing the targ heirs and betraying the mad king had declared for Robert as King. I'm sure he expected to get something in return but that wasn't a guarantee. At this point it woulda been a slog but Roberts alliance of the north, storm lands, the vale and the riverlands was enough to beat house Lannister and the rest of the west if need be

The lannisters joined that war late and joined on the side of what at that point would more likely be the winners. They fell in line so it's not a sure thing that they would have gave Robert( or Ned) trouble in the future but certainly with the lord's daughter as his queen that made that less likely

Stark and Lannister weren't mortal enemies up until Bran got pushed .