Learn English – Origin of the expression “to do someone in”

One of the colloquial definitions of grind is (from the Oxford English Dictionary):

Steady hard work; labour of a monotonous kind, esp. close and hard study; an instance of this, a dull and laborious task.

The first written use of this is:

1851 B. H. Hall Coll. College Words, Grind, an exaction; an oppressive action. Students speak of a very long lesson which they are required to learn, or of anything which is very unpleasant or difficult to perform as a grind.

So the grind of one's day would be the daily, monotonous work that one has to do. It is more common to refer to this as the daily grind.

ODO's entry for slide defines "let something slide" as:

negligently allow something to deteriorate:
Papa had let the business slide after Mama’s death

Therefore, the connotation is quite literal in that you are letting something slide down. Furthermore, there is also an implicit restriction involved in your expression which would expand to something along the lines of:

Let it slide this time

In other words, you're allowing the situation to deteriorate temporarily.

The etymology of the word slide dates the first use of let (something) slide to the 14^th century:

O.E. slidan "move smoothly, glide," from P.Gmc. *slidanan (cf. O.H.G. slito, Ger. Schlitten "sledge"), from PIE root *(s)lei- "slide" (cf. Lith. slystu "to glide, slide," O.C.S. sledu "track," Gk. olisthos "slipperiness," olisthanein "to slip," M.Ir. sloet "slide"). Phrase to let (something) slide is recorded from late 14c.