Make your way down the ramp until you meet up with Ellie. Walk around behind the drilling machine to find a blue panel here, before you reach it, look to the right to find a small hidden passage leading to a power node (40). Return to the panel and use it. Remove the power core out and take it back to where you met up with Ellie. Stick it in the empty slot.
Perform a 180 degree turn so that you are facing away from the drilling machine and walk up the small ramp here. When you reach the top, turn left and use kinesis to grab the diamond semiconductor. Now continue along the path here to the right until you reach the cargo lift. Activate the control panel and ride it across to the far platform where you will be attacked by a pair of super-slashers. Once they are down, use kinesis to remove the kinesis panel near the lift and pick up the circuit board. Return to the lift and ride it back to the previous platform.
Kill the spitter waiting for you here and climb back down the ramp to the drilling machine. Use the action button on the relevant location next to where you placed the power core earlier. A hack panel will now appear next to where you just imputed the circuit board. Hack this and it will power up the drilling machine, but unfortunately will also trigger an attack of the pack. The pack children will come from two sources, a hole in the wall in front of the drilling machine and down the dirt ramp we just came from.
Ellie will have a little trouble controlling the machine at first, so your best bet is to stand at the foot of the ramp where you just returned from otherwise the drill bits will mess Isaac up pretty bad. Once you are set up, we essentially have to defend ourselves against an endless swarm of the pack for a short time until the drilling machine is in position and a platform descends tothe ground. Jump on this and ride it up to the top.
Now we are stuck on a tiny platform, riding through a tonne of baddies, so get ready to use some stasis and some fancy footwork - oh and stay clear of the drill bits or they will slice you into a million pieces. Soon after the drilling machine starts moving, an assortment of three necromorphs either spitters or slashers will climb up onto the platform to have a go at Isaac, dispatch them.
After the first encounter, look to the platform that appears on the right, use kinesis to grab as many of the explosive canisters here that you can and place them on the floor of the platform for when we need them. Soon after this area, another platform on the right will have three lurkers on it. Launch one of the canisters to take them out before they start bombarding you with projectiles. A little further forward, there will be another platform on the right with a group of lurkers. Again use your kinesis to toss an explosive canister at them.
Following this area, spitters and slashers will start climbing up onto the platform from either side with a lot more regularity. So be sure to keep an eye on both sides of the platform to avoid any unnecessary damage. Other than that, use the remaining canister when you get a clear shot on the second group of lurkers to the left (the first group is on a raised platform and might be difficult to hit). Once you have successfully defended the platform for long enough, you will be rewarded with a short cut-scene.
Note: For finishing the drill ride you will earn the achievement/trophy 'Knock, Knock'.