I find whenever I feel the need to use a #region, there's something fishy going on with my code; I see the same in your SendPlejeFamilieObjekt method: all those WriteLine/Flush calls look very suspicious. Why do you flush at every single line written?
Have you considered overriding PlejeFamilie.ToString() to return the equivalent string? You could do _sWriter.WriteLine(pfamilie.ToString(); and have no code to modify in that method the day PlejeFamilie needs a new member.
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The same applies to BiologiskFamilie; whenever something changes in these classes, you have multiple placedplaces that must change. I would refactor this code so as to minimize the amount of code that needs to be written (therefore possible omissions and bugs) when a change becomes required.
Why is everything static? This class has disposable private fields, I think it should be an instance, and should implement IDisposable so it could be used in a using block and then the client code doesn't need to "remember" to call EndConnection.