They would not clone Henriette Lacks, they would just clone her cancer tumor.
Those cells are reproducing continuously because they are cancerous. Find a way to cure them of the cancer, turn them into totipotent cells, and they would develop to a clone of Henriette Lacks, but lacking immortality.
Now, if we can find a way of curing them of the cancer (that is, forcing them to work and convert to liver, bone, skin etc. tissue -- which is precisely what cancer cells don't do -- while not reproducing at an alarming rate), but retaining the capacity to keep dividing without telomere degradation, and we may have something.
Until then, we only have something to study.