Asset inflation ran quite high throughout QE. The same number of dollars might be obtained by selling half of what one purchased with borrowed money at the start of the program.
QE involved the Fed buying assets, not giving away money. What you're saying would suggest that the Fed made massive profits for the public, which is actually correct. Overall QE, apart from it's other effects, has been hugely profitable for the US government. Not so much here in Britain, unfortunately.
The government bought them from the banks, using money the Treasury created. When the assets are sold the created money is 'retired' and any profit returned to the Treasury.
IMHO QE in the US went too far, but it was absolutely a necessary programme that stabilised the US economy at critical moments. The increased reserve requirements on banks, while dragging on bank profits since, were prudent and seem to be working well, especially compared to Europe.