After listening to the clip I can assure you (as sCOSwt does in a comment) that the sound “kay” is not heard and that the sound of the French “R” is sufficiently distinct; however it is not the typical variant (allophone): it is somewhat weak, tending towards the Belgian pronunciation of r or that of the Scottish “ch” as in “Loch Ness”. That is probably why, your friends, not being indigenous speakers of French, believe they are hearing “ê” instead of “êr”.

later addition: Now that I think of it I must add that the slight divergence in the pronunciation of r that I mention above is even more important in the source you provide for « mer » and that you might find other sources that remain faithful to standard French.