Q:

What happens to a losing player's cards in Exploding Kittens?

When a player draws an Exploding Kitten and cannot defuse they are out. Do their cards go back into the draw pile or discard pile?
So if you have a Streaking Kitten and an Exploding Kitten and you pick up a second Exploding Kitten, then you end up discarding 2 Exploding Kittens. So this means there are 2 less Exploding Kittens than number of people remaining.  
So there is no clear winner?

A:

They go back into the Discard Pile.  

EXPLODING KITTEN
  ... Unless you have a Defuse Card, you're dead.
Discard all of your cards, including the Exploding Kitten.

Since there are one more Exploding Kitten Card to start with than normal, there still can be a winner.  

STREAKING KITTENS
  ... 6. Insert enough Exploding Kittens back into the deck so that the total number of Exploding Kittens is equal to the number of players - this is 1 more Exploding Kitten than normal because the Streaking Kitten Card allows a player to hold an Exploding Kitten in their hand without exploding!

Exploding Kittens Rules
Exploding Kittens with Streaking Kitten Rules

A:

According to the rules, you discard the cards, so they go in the discard pile.

You must show this card immediately. Unless you have a Defuse card,
  you’re dead. Discard all of your cards, including the Exploding Kitten.

