Ok, from what I've seen you can just remove the registry entry for the dll. I tried it already and nothing seems to be messed up with the file missing. It's almost as if someone at MS had LDA installed and didn't bother to clean up the registry before publishing the update.
You can paste this into a batch file and run it as administrator or just open an administrator command prompt and paste it into there to quickly get rid of the entry.
echo Start
echo #
echo ######################## Default dirctory for x86 x64 ########################
echo #
echo this command is default system32 directory for x86 OS or x64 OS
cd %windir% & cd system32
reg delete "HKCU\Software\Microsoft\Windows\CurrentVersion\Run" /v "Logitech Download Assistant" /f
reg delete "HKLM\SOFTWARE\Microsoft\Windows\CurrentVersion\Run" /v "Logitech Download Assistant" /f
echo #
echo ######################## Change the dirctory for x64 ########################
echo #
echo this command is x86 application's registry for x64 OS
cd %windir% & cd syswow64
reg delete "HKCU\Software\Microsoft\Windows\CurrentVersion\Run" /v "Logitech Download Assistant" /f
reg delete "HKLM\SOFTWARE\Microsoft\Windows\CurrentVersion\Run" /v "Logitech Download Assistant" /f
echo End
I found the code above here.