The coffee shops in Kuwait have been picking up a lot. So you will definitely be having many options coming up because of the rising number of shops.
Check out the C A F É R E A ❕ coffee festival on 13th January 2018 - 9am to 9pm at Al Safat Square. Most of the coffee shops are taking part, so you will get to check them out and know about them.
Location: Al-Safat Square goo.gl/maps/qPLhum25acT2
This Instagram account is covering all the coffee shops in Kuwait https://www.instagram.com/gahwaa.kw/
Mark wrote a post on Coffee shops and its locations: http://248am.com/mark/food/list-of-speciality-coffeeshops-and-their-locations/
The location vary according to the traffic and timings.
Example: Shuwaikh Coffee in Shuwaikh CDC (Creative Design Center) has a large seating area which would provide you with the comfort for working from there.
Also, you can check out Working Spaces in Kuwait.