Sorry Tom... You don't want to use a totally dead stem. You want to use one that is infected though and showing signs of the disease.
When you cut the stem, products of the xylem and phloem will immediately run out of the plant. When you make your cut you don't want to lose that fluid, so you can either hurry and turn the cuttign upside down to keep the liquid inside the plant, or you can make the cut and immediately let it drip into a cup of water and let it stay in the cup for a bit. Within a half hour at the most you will see the milky liquid form in the glass.
You don't need distilled water. Any water will do.
One of the ways the viruses travel is from cell to cell or through the tiny openings between the cell membranes. Some virus attack a whole plant at once and some first one side then the other. Alot of the viruses move up and down through the plants through the xylem.
As they mov e through the plant they change the sugars the plant takes up to a milky liquid. A bacteria virus can replicate itself to thousands in just a matter of minutes.
If you don't see a milky liquid then your plant could have other issues. You might dig around the base of it and pull up some of the roots in that area. Look to see if any of the roots have fat spots on them. You may have some nematodes.
Though with it dying like that looks more like a bacteria wilt to me. Your plant likes dry conditions. It suffers if the ground is to wet. It can get boytritis, a fungal disease.
With a fungal disease, if you get down by the roots , and scrap away the dirt, some time s you can se e fuzzy lookign stuff. Fungi can come in several different colors so there is no tellign you to look for one specific one. If it is a fungi and can be narrowed down there are several products on the market that might help.
Also, look on the leave s of the branches that are starting to die. Do you see any yellow rings or smal black spots or yellow spots? Look on the back of the leaves, do you see any tiny bumps or little white things sticking to it.
Have you checked in the soil to see if there are any type of bugs or insects there? Also look at night. Some pests only come out at night. Do you have snails and slugs in your country?
