After you approach the hospital, proceed through the open gate to the courtyard. Notice the military helicopter on the left.
Circle around the chopper and get inside the machine. Search the large compartment at the end of the fuselage. Remove some items to access the walkie-talkie in the right lower corner. Pick it up.
Leave the helicopter and head for the hospital. Once Kate hides behind a pillar, use the walkie-talkie on the soldiers at the front office (achievement Colonel, are you receiving me? Over.). After the men leave, go through the corridor where you'll find the rooms of doctor Zamiatin and doctor Olga (left of the front office, looking from the entrance).
After you talk to Zamiatin (Cassandra achievement - you can get it if, after reading Dr. Olga's mails, you tried one more time to convince the hospital's director about Efimova's real identity) and Steiner, head for Olga's room. There, you'll find Kurk. Unfortunately, your Youkol guide will be unconscious - hypnotized with a pendulum and drugged - you'll have to deal with both of those problematic conditions.
Zoom in and notice the note attached to the left side of the Kurk's seat. Take the clip that's holding it. Notice something like a small tube hanging out from the headrest. If you click it, Kate will try to wake Kurk, to no effect, unfortunately.
Zoom in on the back of the seat and slide down (with a mouse movement) the headrest cover. You'll see little cogs that move correspondingly to the pendulum - block them (upper active point) with the clip you took from the note. Now look at the other side of the seat.
Zoom in on the instruments you'll see on that side of the seat and displace the needle to the right (mouse movement), to cause the fluid to leak from the syringe. Now put the needle back on its place. Pull back (move the mouse up) the piston of the syringe and fill it with the potion you got from the shaman. Plug the syringe and pull down the switch on the right.
Once your guide comes back to his senses, you'll just have to undo the restraints on his forearms, holding him on the seat. Below the right side of the seat, you'll find a small flap, which you'll have to pull down (with a mouse movement) - underneath, you'll notice a keypad, most likely requiring a key code, which you do not know.
Zoom out and look around the room in search of something that may put Kate on the right track. Approach the desk and pick up Olga's figurine, standing left of the monitor, along with an underlying note - message from doctor Mangoling. Read it (documents in your inventory) - unfortunately, the fragment with the key code is missing. Having no other choice, zoom in back to the keypad and use the figurine on the active point right above the keypad (achievement The long goodbye).