Any parent of a young type-1 diabetic would have some sort of high sugar item (candy, juice, glucose tablets, etc.) to bring their child out of a hypoglycemic attack. Yet Jeremy's mother ran into the street carrying him yelling for help like she had no idea what was happening. Her first thought should have been to a diabetes-related issue.
The child's problem is stated as diabetic shock, caused by injecting water instead of insulin. Diabetic shock is hypoglycemia, or low blood sugar, which is caused by too much insulin, not too little. It can happen quickly. Too little insulin causes hyperglycemia, or high blood sugar. This can cause diabetic coma, but it takes days or weeks to develop. The mother of a diabetic child would have been monitoring his blood sugar level and would have noticed the increasing blood sugar level and symptoms long before they resulted in coma.
The gentleman from the charity gives Frank the gift of a pen, which Frank accepts. However, in the second season, Curtis Swint tries to give Frank a gift of a bottle of scotch but Frank refuses the gift, and states that the Police Commissioner is not permitted to accept gifts (gratuities). Even worse, the gift is given after Frank details the various crimes that he may be charged with, which would be a conflict of interest.