Unfortunate you had a bad reaction to the insulin, sorry to hear of your struggles.

Sodium hydroxide would be added to adjust the pH of the buffer used.

That product (Tresiba) consists of the insulin (insulin degludec, which is human insulin with one amino acid removed (Threonine 30), and a long chain fatty acid (hexadecanedioic acid) conjugated to the Lysine residue immediately preceding that removed threonine (Lys29)). In addition to what ever amount of insulin they put (based on dosing) they have a variety of inactive ingredients necessary for stability of the protein, they are:

Inactive ingredients for the 100 units/mL are: glycerol 19.6 mg/mL, phenol 1.50 mg/mL, metacresol 1.72 mg/mL, zinc 32.7 mcg/mL and water for injection. [1]

This insulin form is stored at a near neutral pH 7.6, the various components listed above can each influence the overall pH of the solution so as such to adjust the pH to 7.6, sodium hydroxide (a strong base which in solution exists only as Na+ ions and OH- ions) can be added.

The pH of insulin is quite important and storage at other non-physiological pH's can result in changes to the oligomeric state of the protein (IE it exists as a hexamer... 6 insulins together in a complex). If the pH gets too basic then it falls apart into monomeric subunits, if it gets to acidic (low pH) then it falls apart but into a totally different conformation where two insulins stack inverted to one another (this is a dimer). [2]

Does this answer your question? Sodium Hydroxide is used to adjust the buffer pH to a neutral range to ensure the appropriate oligomeric state of the insulin necessary for in vivo efficacy.

Mind you, I don't work on insulin, I don't work for this company, and I've not read much about this protein. This is the best I can do for you with a few mins of research into it.

The FDA has a pretty comprehensive document on this insulin form:

https://www.accessdata.fda.gov/drugsatfda_docs/label/2015/203314lbl.pdf

If you want to know more about the oligomeric state and pH dependence of the protein you can look at:

http://pubs.rsc.org/en/content/getauthorversionpdf/C6AY01573E

Both of those two documents were referenced for me to answer your question.