Visas for South Korea are not required for stays of less than 90 days. A visa is required to enter North Korea for any length of time.
South Korea Timeline
A visa for South Korea can be obtained in approximately four business days and can be expedited for an additional fee.
North Korea Timeline
North Korea does not have an embassy in the United States, so all visa requirements must be sent to the United Nations in New York. As a result, getting a visa for North Korea can take weeks.
A valid passport is required to visit both countries. The passport needs to be valid for at least six months past the date of departure. It can take four to six weeks to get a new passport.
You need to have the proper type of visa (for example, business, student, tourist) before entering either country. The visa cannot be switched in-country and changing visas before departure can delay processing time.