The underlying issue is that the Kings are breaking the puck out of their own zone like a rec-league rollerhockey team. They've lost the structure that has characterized their defense all year and especially last year; as a result, they can't generate much speed through the neutral zone, and thus they don't attack with speed. This is a big reason why the forwards are underperforming. The Kings are playing like they're constantly on the penalty kill, chipping every puck off the glass, trying to generate offense with quick rushes, and conceding a TON of offensive-zone time to the Blues.
I think there are several causes: first, we're REALLY missing Mitchell, and Regher is not even close to a replacement for him. Regher is not a particularly effective player at all. He's physical, but his first passes are poor at best. This is why he's an adequate penalty killer, but a liability at even strength. He's holding Dughty back when they're paired together because Doughty needs to do all the work breaking the puck out. Muzzin and Ellerby are similarly ineffective at breaking the puck out. Second, the forwards need to do a better job in the defensive zone. They need to play a bit deeper and win the board battles you mentioned. Doing so would also make them more available for breakout passes. This will probably concede a bit more space to Blues' defensemen to shoot the puck, but if the Kings are winning board battles and breaking the puck out better, then that's a good trade. The forwards also need to backcheck harder through the neutral zone, on those occasions when the Blues are counter-attacking. Indeed, third, the forwards and the defensemen need to pressure more up-ice. The defense's gap control has been horrible, and that allows the Blues to enter the zone with impunity. They don't have to fight to get the puck into the zone.
Really, the starkest difference in all three games is the way that the Blues play as a tight, five-man unit that moves up and down the ice together. They support each other extremely well in all three zones. Every time we start to get a cycle going--our bread and butter, the thing that wins us games--the Blues pressure effectively and then move the puck up the ice easily because they always have an extra option for a breakout pass. ALWAYS. This happens consistenly: two of our forwards chase down a puck below the Blues' goal line, get into a board battle with two Blues, the puck comes free, and two or three other Blues have passing options that our remaining forward cannot stop. Thus, they come out of their zone quickly and efficiently. We are sometimes able to contain them when our defenseman pinches down the boards, but if you notice, the Blues almost always try to bring the puck up through the middle, which their compact structure allows them to do. Our defensemen are not controlling the gaps well--in part because the Blues play so deeply on defense--and thus they are always attacking with speed through the middle. It's absolutely fundamentally sound hockey, kind of amazing to watch, really.
The Kings, on the other hand, are spread out, not working as a five-man unit. Our defensemen are outnumbered by three Blues on the attack, and our forwards are not available for breakout passes. Thus, when we are able to gain possession in our own zone, we can't get it out, and Blues regain possession, usually along the boards or at the blue line, and their attack continues. The Blues outnumber us at the puck in every single zone, on every singe shift.
It's a credit to Quick that we are in this series at all. As Herby said, he is the real franchise player, giving us a chance to win even when we are being outplayed in every aspect of the game. If the Kings are going to really make a series out of this, they have to play as a five-man unit. They did that for period one of game two, and they have been not terrible for a few other stretches. If they can't do that for sixty minutes, then our season is over.