So I saved this woman's ass from getting the dagger in the back by one of the undercover forsworn men. Come to find out that shes General tullius agents. As a staunch Stormcloak right after I heard that I decided to conjured my duel swords and give her a good dose of slicing and dicing. I end up getting a bounty over my head and my stormcloak brothers and sisters starts attacking me. I eventually paid off my bounty... but still it pisses me off that I wasn't able to tell the guards that she was an imperial Legion spy.

It's illegal to murder people, even if they are spies.

war is war so it's not illegal... especially since the Jarl sides with the stormcloaks anyways.

you are the only one who knows she's a spy, so to their eyes you are just another murderer to arrest

During a war laws and certainly mores do not suddenly get suspended.
You cannot expect a city to waylay its laws just because someone is accused of being a spy.

Would the Jarl even care that she's an Imperial Legion spy, given that he's a Legion sympathiser himself and the Stormcloak-supporting Silver-Bloods are a major thorn up his [censored]?

I don't know which side you're playing on http://elderscrolls.wikia.com/wiki/Thongvor_Silver-Blood. In fact he's an Ulfrics supporter.

It wouldn't look good for him to just allow an imperial legion spy roaming freely in his city.