Head on down the corridor in front of you. Just go forward toward the elevators that are at the end of the hallway you find yourself in.
At the end of the corridor you have 2 Guards that you need to deal with. You need to watch them and counter manually as you have nothing to telegraph the incoming attack. Just watch their movements and it will not be too hard to take them down. As you deal with the first 2, a second pair will approach you from the the right. Deal with them and head for the elevators.
When you get to the lip of the next elevator entrance, wait for the Guard to turn his back then take him down. Now you need to fight the rest of the guards. Be ready to make use of the Human Shield tactic throughout the rest of this section.
head over to the left and into the Conference Room. Clear out the 2 Guards that come in after you. Head on through to the Atrium. Move in and take out the 2 guards you see on the same level as you. There are 3 more guards to the right along the walkway.
Deal with them and head on through the broken window. Go on through the first door on the left to finish some of this up. Daniel Cross is waiting inside the Animus room with a gun and he is trying to take down Desmond. Cross will suffer an Episode of the Bleeding Effect and be somewhat withdrawn for a while. Have Desmond chase after him.
If you take a moment to use Eagle Vision in this room you will find the messages that Subject 16 left all over. Now chase after Cross as he runs. You will encounter a few guards that you can either ignore or engage. Either way, chase after Cross and head on through the window after he shoots it out. Follow him on through as he runs through the rafters of the Animus Lab. After that you just need to run him down and kill him. That will finish this.
Desmond has a gun now for this final stretch. You can use it or you can just attack them with the hidden blade and knife combo. Head on down the hallway and over to the left. Deal with the guards that come after you. Deal with the group of Guards and a Secretary will let you into Vidic's Office. Time to get on through a cutscene and then you need to finish things out here. William will grab the final power source from the desk as Desmond grabs the Apple.
Use the Apple to make it out of the Abstergo Facility without any further trouble.
Placing the Power Source
Once back inside the temple, head on up the stairs to the right of the Animus set up. Head on up the stairs, swing across the gap and make your way up and to the right. Climb on up the pillars and make your way over to the other side. Head over to the right to the next gap. Look down to see 2 poles you can use to swing across. Do so and climb up the far side. Head over to the right to trigger another cutscene.
Jump on over to this after the cutscene and head over to the right using the various handholds. Head over to the left after that and you will find the insert for the final power source. After that it is time to head back into the Animus.