It's usually not very difficult to insert the keyboard ribbon cable. The hardest part is usually getting a grip on the ribbon cable itself, or seeing exactly where to slide it in. The one in the photo, however is a little more difficult since there isn't a "tab" on the ribbon to "angle" and pull or push the end of the ribbon cable into the connector. If you're a little older like I am ( early 40's), reading glasses are a big help. I got the most powerful common kind of $10 reading glasses I could find (I only use them for computer repair or looking at fine details on my iPhone 5). They have a "power" of +3.00. Reading glasses (cheap ones) usually come in +1.00, +1.25, +1.5, ... up to +3.00 power (strength/magnification).
After looking closer at the photo, and looking at the same connector on one of my 15" MacBook Pros (they look the same as the 13" MacBook Pros), it looks like the retaining flap is in the down/locked position. This flap is very short (thin) and runs the entire length of the connector on the opposite side of where the ribbon cable is inserted (in this photo, it's above the connector toward the top of the photo). It needs to be flipped up to insert the ribbon cable, then pushed down to lock it into place; hold the ribbon cable in.
In this case without a "tab" on the ribbon itself, you can use a pencil eraser to push the ribbon cable in, or your finger (if your finger is too dry and doesn't have any friction to push the ribbon cable in, it won't be able to get a grip). The ribbon cable slides into the connector very easily with very little friction, as long as the retaining flap on up on the other side of the connector. If the flap stays down when you're sliding in the ribbon cable, it won't go in far enough. Again, in this photo, it actually appears that the flap is in the down/locked position.
Sometimes, you can still see the traces (copper conductive paths inside the plastic ribbon) going into the connector, but barely. If too much is visible, as it would appear in the photo, then it wasn't inserted enough. I would suspect that you didn't release the retaining flap on the other side of the connector first.
Use a spudger to lift this retaining flap. If you don't have a spudger, your finger nail should easily get underneath it enough to lift it up. If you try sliding the ribbon cable in after lifting the retaining flap and it goes in farther, then you've got your answer!
The fact that your charger and battery LEDs are responsive is a good sign that your logic board is still OK. (Though, not conclusive!) The Battery and MagSafe connector are connected to opposite sides of the logic board.