Reaching the opposite end of the shaft shouldn't take too much of your precious time. Once you're outside use your hook and grapple up to an upper ledge found directly in front of you. Eventually you'll find your way to Arkham East. Watch a new cut-scene and glide towards the ground level. Talk to Louie Green there (one of the guards).
Take your time to look around, however you'll soon receive a warning that your batmobile is in danger. Familiarize yourself with your current goal and don't forget to check your personal map in order to find out where you must go next. As you've probably noticed, you'll be moving north-west, towards the Arkham North area. Approach a large gate and enter a short tunnel. Stop for a second and listen to a new radio transmission.
Get closer to an ambulance. Activate detective mode to locate two armed thugs and wait for them to turn around. Once this has occurred use your hook to reach an upper ledge quickly. Make a few steps forward, drop down and perform silent takedowns on both inmates. Don't forget to keep using the crouch key while trying to surprise them.
You may now head through a large gate leading to Arkham North. Start moving towards your batmobile, however I would recommend that you get on top of one of nearby structures. Doing this will allow you to surprise inmates surrounding your vehicle by using a Glide Kick ability. Deal with the rest using standard attack methods. Use counterattacks to prevent Batman from losing health.
Once you've defeated all hostile units approach the trunk area of your car and press the action key to add Explosive Gel to your inventory. Listen to what Batman has to say and turn on your detective mode. Search the area in the vicinity of your batmobile to find a pipe. Scan this object to initiate a new investigation.
Once again you'll have to follow the trail, however this time DNA samples are going to be easier to spot. Eventually you'll have to travel to a gate leading to Arkham West, however upon your arrival you'll find out that the entrance is locked.
Proceed to your right and make sure that you're still using your detective mode. Find a weak section of the wall (screenshot #1), choose the explosive gel from your inventory and apply the gel by pressing the action key. You must now detonate the gel, unlocking a nearby passageway. Find a door leading to Arkham West.
You can now keep following the trail closely or you can proceed directly to your destination which is going to be the Medical Facility building. The entrance to the building is being guarded by a large group of thugs and obviously your objective will be to get rid of them. It shouldn't be a problem as long as you don't forget to use counterattacks.
Open the main door leading to the Medical Facility building and you'll find yourself standing inside a zone marked as the Medical Foyer. Make a few steps forward to trigger a cut-scene with Harley Quinn. You can't do anything from here, so head back outside to the main area of Arkham West.
Once you're back outside use the hook to grapple up to the roof of this building. Head forward and turn left. Eventually you should be able to locate a weak section of the wall (screenshot #2). Use Explosive Gel to blow up the wall and head inside. Open a small door in order to enter the Medical Facility building (Maintenance Access area to be more precise).