Danish Rye Bread Starter Problems

I'm not familiar with the recipe, but with others that are similar. The mixture is meant to try to capture wild yeasts and to encourage the growth of any natural yeasts which are normally present on whole grains like rye. You would want to have very fresh rye flour, preferably a dark whole grain variety, to start with. Yeast needs air, so bowl covering should be done using cheesecloth or a tea towel, not plastic wrap or a lid. It may also take several days to get one going, sometimes up to a week or more, depending on how warm conditions are and the amount of wild yeasts floating in the air around you. I bake quite regularly, so if you did this in my kitchen, you would probably get a starter in just a few days. But if you don't bake with yeast much, our modern clean homes generally don't have much floating around in them. Not yeast, anyway. You may be able to still use your current mix, though it would need enough buttermilk to make it a rather thick batter consistency instead of a thick dough. Another option might be to seek out a bottle conditioned beer, and use that as the liquid. Those undergo a second fermentation in the bottle, so contain residues of live yeast in the sediment found in the bottom of the bottle. I like to use either Chimay Red or a Sierra Nevada Pale Ale when I make bread sometimes, and both of those are bottle conditioned. Both make great starters. But mostly, what you need to start a sourdough is patience and warmth.

A stiff starter doesn't necessarily jump out as a problem, although I wonder if whole (dark) or light rye four were specified. The dark/whole absorbs more liquid due to the bran, so it would appear stiffer. A quick Google search for "Trini Hahnemann’s danish rye" yields a recipe that has 200g of rye flour and 350ml of buttermilk in the starter. It looks mighty tasty. Where are you sourcing your recipe?

Here is how I go about rye bread. I hope it helps you. It is one of my favorite breads, and makes amazing toast.
https://thesolitarycook.wordpress.com/2012/03/16/preferment-for-rye-bread/
https://thesolitarycook.wordpress.com/2012/03/18/rye-bread-with-dill/