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 was 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.