I started working on this question after it was posted to MathOverflow and found bounds similar to those found by Justin Gilmer: upper asymptotic density of the happy numbers 0.1962 or greater, lower asymptotic density no more than 0.1217.  However, I was also able to prove that the upper asymptotic density of the happy numbers was no more than 0.38; Gilmer mentioned in his paper that the question of whether the upper asymptotic density was less than 1 was still open.

A writeup of the result is at <http://djm.cc/dmoews/happy.zip>.  The method used to find an upper bound on the upper asymptotic density is to start with a random number with decimal expansion  $??\dots{}??\hbox{\#}\hbox{\#}\dots{}\hbox{\#}\hbox{\#}$, where the digits \# are independent and uniformly distributed, and the digits ? are arbitrarily distributed and may depend on each other, but are independent of the \#s.  Then if there are $n$ \#s, asymptotic normality implies that after applying $s$, we get a mixture of translates of a distribution which is approximately
normal, with mean $28.5n$ and standard deviation proportional
to $\sqrt{n}$.  If $10^{n'}/\sqrt{n}$ is sufficiently small, each translate
of this normal distribution will have its last $n'$ digits approximately
uniformly distributed, so we get a random number which can be approximated by the same form of decimal expansion we started with, $??\dots{}??\hbox{\#}\hbox{\#}\dots{}\hbox{\#}\hbox{\#}$, where now there are $n'$ digits \#.  Repeating this eventually brings us to numbers small enough to fit on a computer.

The method used to find the bounds similar to Gilmer's was to start with a random number of the form  $dd\dots{}dd??\dots{}??\hbox{\#}\hbox{\#}\dots{}\hbox{\#}\hbox{\#}$, where the ?s and \#s are as before, the $d$s are fixed digits, and there are approximately the same number of $d$s and \#s, but very few ?s.  Then if the parameters are appropriately chosen, we can show that after applying $s$, we again get a random number which can be approximated by the same form of decimal expansion, $dd\dots{}dd??\dots{}??\hbox{\#}\hbox{\#}\dots{}\hbox{\#}\hbox{\#}$, and repeat this step until the number is small.