Learn Italian: Non vedo nessun senso in questo compromesso.

Because senso is a masculine singular noun starting with a single consonant (an s followed by a vowel). Nessuno follows the exact same rules as the indefinite article un/uno. Since you would say un senso, you use nessun senso. You only use nessuno before masculine words that take uno, such as those starting with z or s plus another consonant (like nessuno specchio).

Why are there two negatives (Non and nessun) in this sentence?

Italian requires a double negative when a negative word comes after the verb. Since nessun (meaning "no" or "any" here) comes after the verb vedo, you must place non before the verb. While English says "I see no sense," Italian literally says "I do not see no sense."

Could I change the word order to avoid the double negative?

Not naturally in this case. While it is true that putting a negative word before the verb lets you drop the non, that rule is mostly for when the negative word is the subject (like Nessuno vede... meaning "Nobody sees..."). Here, nessun senso is the object of what you see, so it belongs after the verb, requiring the double negative structure.

Do I need an article before senso?