IL
1) Get a hitchayvut for the emergency room
1a. Ask the receptionist if there is a doctor with appointments in any of the six clinics in walking distance who is willing to see you and agree that this needs stitches
1b. If there isn't one, check to see if this is the day that the Chutznik doctor who is a friend of yours's secretary has hours, so you can phone him and ask his secretary to issue the hitchayvut.
1c. If you succeed at 1a or 1b, go to the emergency room. If not, go to 2).
2) Call the kupah's hotline and ask for a hitchayvut to the emergency room. They will tell you to wait for the next morning and go to 3). If you cry to them ("It really hurts and I can't take care of my baby like this!"), they will issue the hitchayvut most of the time. Total cost: 42 shekel. If they do not issue the hitchayvut, go to 3).
3) If it isn't during the five hours per day that they are closed, go to your kupah's extended hours clinic. Wait anywhere between twenty minutes and three hours to be seen and stitched up. Total cost: nothing or 18 shekel or 36 shekel or 42 shekel, depending on how it is filed (not your choice).
Please note: this should not be used as an accurate description regarding what you should do for a real-life situation, as inaccuracies may likely be in there due to my not personally having dealt with stitches (DW chose not to bother me in yeshiva). See, for example, LongTimeLurker's comment below.