This link (which is posted in the comments) covers the question:
Apparently the thermosphere exhibits more variation due to a host of factors. These include season, location on Earth, solar activity, and so on. Because of that, anyone applying a single formula for this must realize that it might be up to an order of magnitude off. However, the density profile varies by many orders of magnitude, so I still think it's meaningful to talk about.
Here is a sample output.
km O/cm3 N2/cm3 g/cm3
1 2 3 4
0.0 0.000E+00 2.120E+19 1.304E-03
50.0 0.000E+00 1.361E+16 8.373E-07
100.0 3.995E+11 8.467E+12 5.173E-10
150.0 1.907E+10 3.236E+10 2.190E-12
200.0 4.918E+09 3.538E+09 3.100E-13
250.0 1.696E+09 5.676E+08 7.348E-14
300.0 6.293E+08 1.009E+08 2.181E-14
350.0 2.402E+08 1.873E+07 7.388E-15
400.0 9.331E+07 3.582E+06 2.723E-15
450.0 3.679E+07 7.029E+05 1.065E-15
500.0 1.471E+07 1.413E+05 4.394E-16
550.0 5.959E+06 2.907E+04 1.926E-16
600.0 2.446E+06 6.118E+03 9.136E-17
650.0 1.017E+06 1.317E+03 4.793E-17
700.0 4.279E+05 2.896E+02 2.815E-17
750.0 1.823E+05 6.506E+01 1.840E-17
800.0 7.861E+04 1.492E+01 1.308E-17
850.0 3.429E+04 3.494E+00 9.863E-18
900.0 1.513E+04 8.346E-01 7.717E-18
950.0 6.750E+03 2.033E-01 6.177E-18
1000.0 3.045E+03 5.048E-02 5.015E-18
It's also meaningful to note that the atmosphere becomes highly differentiated at high altitudes. So basically, the other elements fall to nearer to the surface and almost only Hydrogen is left at super high altitudes. This is what the numerical models spend a lot of their time on.
Here is a plot of the density.
Note that I had to give different units of kg/m3 here. Just because Excel would muck up the formatting with smaller numbers.
It's actually rather interesting that an exponential trend just won't fit this data. A power fit makes a decent approximation. I don't know why. I can't explain why the atmosphere would better fit a power law than an exponential trend, since the exponential trend comes from the ideal gas state equation and the force balance. That's actually a rather intriguing statement. Of course, even the power law isn't fantastic, and I'm sure a point or two are off by a factor of 2.
Nevertheless, this is usable for the query in the question. The density figure could be combined with orbital mechanics to ballpark the orbit's lifetime.