Just finished Assignment Ada. It gets pretty tough at some parts. I died about 8 times throughout the whole level. The only tips I can really give is snipe when you get the chance and watch your back or you will get stabbed in it. Oh and use those grenades well because there will be mobs of infected coming at you.
But I don't think you have to beat Assignment Ada to unlock the Chicago Typewriter for the main game. When I beat Separate Ways it told me I could now buy it from the merchant. And later, when I beat Assignment Ada it told me to start a new game from the cleared Separate Ways file to have the Chicago Typewriter available.