I would not say that the Valar envy mortality. Does Eru envy mortality too? Why? Just because they're immortal? The Valar have been around longer than Arda itself, and will continue to be forever.
The reason elves struggled with their immortality because they were unable to keep the world (Middle-earth) around them from changing (the way they do). In fact, I'm not even sure we're given any hints about an elf in the Undying Lands complaining about immortality - I see no reason for one to, aside from scars from his/her Middle-earth past.
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...take counsel with thyself, and remember who and what thou art.
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