Over the Wall
Speak to Elena and she'll tell you that the mourners took her equipment. She needs the distillator that the mourners took, and are holding in the Mourner HQ in West Ardougne.
Offer to try to find it, and she'll tell you that the mourners found and filled in the hole to the Ardougne sewers, and that they need another way over the wall. She directs you to Jerico, who lives two houses north of the chapel in East Ardougne.
He'll tell you his messenger pigeons keep him in touch with West Ardougne, and that he has some friends on both sides of the wall ready to help with a rope ladder. After talking with Jerico, open the cupboard and take some bird feed. Then, go out and take a pigeon cage from behind his house.
Omart is waiting for you at the south end of the wall. He tells you they are ready, but they can't risk it with the Mourners' watchtower being so close. Go to the big gate west of the East Ardougne Castle. Go to the corner of the fences, and use the birdseed on the fence of the watchtower.
Open the bird cage and the pigeons will fly out and distract the mourners. Run to Omart and talk to him. He will call over to Kilron, and they will set up the rope ladder for you to climb over.
Exit the fence, and head around to the front of the HQ. Try opening the door, and a mourner will tell you that several mourners are ill with food poisoning, and they're waiting for a doctor.
Go to Nurse Sarah, located in the larger building south-west of the West Ardougne church (marked in red on the above map). Open the box at the north side of the building, and search it to find a doctors' gown.
Return to the Mourners' HQ, and while wearing the doctors' gown, they'll let you in. Go up the stairs to the east, and kill one of the level 13 mourners there. They will drop a key; use it on the gate. Search the second crate from the east on the north side to find Elena's distillator.
Exit the Mourner HQ, and return to the south-east corner of West Ardougne and talk to Kilron to go over the wall. Return to Elena.
Elena will try to do tests on the plague sample, but it doesn't appear to work. She'll give you three vials and the plague sample. You must bring them to the Chemist in Rimmington.
It is HIGHLY recommended to bank the samples right away! While you have the samples in your inventory, DO NOT teleport via any means except Fairy Rings. DO NOT enter combat. If you teleport or enter combat, the samples WILL BREAK, and you will have to start again at Elena.
Once you bank the samples, you can then teleport and pick up the vials at Falador or Draynor bank, and continue to Rimmington. The Chemists' house is located west of the Rimmington general store.
Talk to the Chemist (do NOT mention anything about the plague sample). Tell him you're a friend of Elena. Tell him you need some touch paper for a guy called Guidor. He'll tell you that they're doing spot checks in Varrock too, and that you wouldn't be safe bringing the three vials with you.
He'll suggest giving the vials to his errand boys, that are just outside near the fire. Follow the below table on who to give what.
Go to Varrock. If you don't have the Priest gown pieces yet, you can get them from Thessalia's Fine Clothes Shop in Varrock. Once you have the robes, head to south-east Varrock, and enter the gate east of Aubury's Rune Shop.
The guard will stop and search you, but will let you pass. Go to the Dancing Donkey bar nearby and talk to the three to get your vials back. Make sure you're wearing the priest gowns, then go into the south-west most building and go through the second door to Guidor.
Tell him that you need help with the plague. Tell him Elena sent you. Give him the sample and vials, and he'll get to work. He will get the same results as Elena. There is no plague!
Return to Elena with the news. She tells you to talk to the King of East Ardougne. Head into the castle to the south, and climb up one of the stair cases. Go into the central room and speak to King Lathas. He'll admit to it, and tell you the reason why, about West Ardougne's King, Tyras.