If you were to write out kilometers (kilometres), you would include the s, and when you say the quantity, you would say the s, but you do not normally add an s to abbreviations like these. So, you will have a plural quantity (these are count nouns), because you would count each kilometer or kilogram as an individual thing.
You would say 1900 km of coastline were affected.
Hmm. I'm reading eb's version, and I agree that 1900 kg is a lot to lift, and 19 hours is too long to wait.
However, I see 1900 kilometers as the subject of the sentence, and of coast/coastline as a prepositional phrase. So, for me, 1900 km were affected. However, if we changed the subject, it might be An undetermined amount of the coastline was affected.