You can do this in one fell swoop with Text-to-columns (dates which are Excel dates are left untouched):
Select the dates (in a single column) that you want to convert, go to data|Text-to-columns, in step 1 choose Delimited, in step 2, untick all tick boxes, in step 3 choose Date and in the drop down next to that choose MDY (this tells Excel how the original data is laid out), then click Finish.
QED.
As an aside, this mixing of dates and text often occurs in Excel when you have copy/pasted a column of dates from elsewhere (usually text) say from a web site, this is when Excel's 'helpfulness' is a pain; it tries to help by converting what it thinks are dates into Excel dates, but how it does this depends on your Locale setting in Excel/Windows, so if you're in the US and it sees 3/4/2015 it converts it into March 4th 2015, had you been in the UK it translates to April 3rd 2015. Wherever you are, if it sees a value above 12 where it expects to see a month number, it leaves it as text.
The mere fact that it has failed to convert some dates implies that it is likely that some other dates will have also been incorrectly converted (the day/month will be transposed).
The way to stop Excel being helpful is to format the destination cells to Text before pasting anything in them, then you do a text-to-column for the whole column as above.