What is the legitimate Islamic way to deal with a string that has knots found in the workplace, after these knots have been untied, especially with the feeling of symptoms such as constriction in the heart, shortness of breath, and the imminent obtainment of a work privilege?

What your colleague did by reciting the Mu'awwidhatayn on the string and untying the knots in it is the correct action, and by doing so, the magic will be nullified, Allah willing. It is recommended to maintain the morning and evening remembrances, and all the reported remembrances as a protection against magic. Also, ruqyah performed by the afflicted person upon themselves, or by a pious and righteous raqi, is a good thing.