Weekends are awesome because I can continue to sleep all day long.
This is like 続く except for now you're the one doing the continuing. Unfortunately, there's no good way to differentiate the two "continues" using just the meaning, but you can think of it this way: With ける, you really care about continuing this thing so you are continuing it (doing the action of continuing). With く you're just like "that's coo' if it continues, but I don't really care...
The reading is the same as 続く, so be sure to know that so you can know this.