I'm assuming that the Chiswell street is the one in central London. With a postcode of EC1Y 4SA or similar
The best option would be the Thameslink rail service from Gatwick to Farringdon station, and then transfer to the underground for a short journey to Moorgate or Barbican, depending on which end of the street you want. There will then be a 3 or 4 min walk.
You can get timetable information here, and the line you are looking for is the Brighton Bedford route.
Then this link gives an idea of the locality (it references a restaurant but you'll get the idea)