Why does "going to kip" mean "going to sleep"?

Solution 1:

There is some more detail on World Wide Words.

The Irish usage as brothel is first recorded in Oliver Goldsmith’s The Vicar of Wakefield:

... to assist at tattering a kip as the phrase was, when we had a mind for a frolic.

The phrase tattering a kip meaning "wrecking a brothel".

The word then came to be used for lodging-houses and finally to refer to the act of sleeping itself.

In Partridge's Dictionary of Common Slang there's also mention of Danish kippe and a possible link to Romany kipsi 'a basket' and kitchema 'an inn' and a possible nautical origin of kip meaning 'hammock'. However, unfortunately I cannot reproduce the complete entry here.

Solution 2:

I can tell you in Scotland the word kip also means bed. The derogatory phrase "Still in his kip" means still in bed (not still asleep), when the person should have been up and about already.