Well it doesn't sound like your S4 is completely dead, at least you can see that Samsung startup blurb. Can you start up your phone into its Recovery Mode (start it while simultaneously holding down the Volume Up, the Home, and the Power buttons) or its Download Mode (start it while simultaneously holding down the Volume Down, the Home, and the Power buttons)? (...note that it won't matter if you run the Odin utility off a PC if you can't get into Download Mode anyway.)
If you can at least get into Recovery Mode, an all-text user interface, try using the indicated keys to navigate to the wipe system cache partition. Wipe that cache and try to reboot your S4 normally to see if it boots up correctly again.
If that doesn't work out, try booting your S4 into its Safe Mode. You need to shut it off and then press the Power button like you usually would but as soon as you see that Samsung logo press and hold just the Volume Down button. If your S4 boots up into its Safe Mode, than that's indicator there's some app installed recently that's causing a nasty boot up problem.
Now if there's still a problem booting up your phone normally, a last resort step is to re-flash the stock ROM. Just note that ROMs are model specific and are not interchangeable. Also this doesn't require you to root your phone (custom ROMs require you to root your phone along with installing a custom Recovery, Samsung stock ROMs do not.) What model S4 do you have and which carrier?
(As a side note, something like sealing your phone in bag with a desiccant only applies to situations when you're trying to dry out your phone after it's been dropped into water. It's not going to fix something like a boot-up issue.)