Without sampling a large population of your nvarchar date data, I will assume the example you gave will be sufficient.
Here I simulate your scenario.
declare @TestDateValue nvarchar(50)
set @TestDateValue = '01/01/05'
select cast(@TestDateValue as datetime) as NewTestDateTime
The variable @TestDateValue is now a datetime. If milliseconds are not necessary, you could also use:
select cast(@TestDateValue as smalldatetime) as NewTestDateTime
I hope this helps.