There are a few things that popped in my mind seeing this:

Generalization

The method can be used to split any enumerable/collection/array, but this is just a minor point.

EnsureValid

IMO this method does a little too much. It: checks if the parameters are valid (desiredLength != 0, which I think should also be desiredLength > 0 given the context) and also takes action to prevent the application from going on (throws exceptions, in this case). I'd change it into something like the following:

private static bool IsValid(string value, int desiredLength, bool strict)
{
    return value != null
        && !(value.Length == 0 && desiredLength != 0)
        && !(value.Length != 0 && desiredLength < 1)
        && !(strict && (value.Length % desiredLength != 0));
}

and let the management of special cases to the caller. In this way, the first line of the Split method becomes something like the following:

public static string[] Split(this string value, int desiredLength, bool strict = false)
{
    if(!IsValid(value, desiredLength, strict))
    {
        // manage the non valid case here
    }

    // other logic
}

Split method

In the current case, for strings I mean, the method seems good. In case you consider changing it for the general case, I'd suggest the following logic:

public static T[] Split<T>(this T[] value, int desiredLength, bool strict = false)
{
    if(!IsValid(value, desiredLength, strict))
    {
        // manage invalid case
    }

    var numberOfItems = (int)Math.Ceiling((double)value.Length / (double)desiredLength);
    var result = new T[numberOfItems];
    var chunkItems = new List<T>();

    for(int i = 0; i < numberOfItems; i++)
    {
        int chunkFirstIndex = i * desiredLength;

        for(int j = 0, currentIndex = chunkFirstIndex; j < desiredLength && currentIndex < value.Length; j++, currentIndex++)
        {
            chunkItems.Add(value[currentIndex]);
        }

        result[i] = chunkItems.ToArray();
        chunkItems.Clear();
    }

    return result;
}

Other than these things, everything else looks fine.

P.S: The code I wrote is not tested, it's there just to better explain the idea.