You may have to manually replace that file. It is in one of the cabs on your ME CD. I find I have it in two places . . . C:\Windows\System and in \Program Files\Common etc. under Adaptec.
WinMe will not normally replace system files that are "present for duty."
What I could do is extract a good copy ofthat file from the CD, then use my old Win98 DOS boot disk . . . drill to C:\Windows\System and copy the new one right over the old one in DOS. There are some repairs to Windows that can't be done when Windows is active.
It could also be a registry problem where some entry has the wrong location in a link to that file. If this just started, use system restore to go back to a date before that problem arose.
FYI, wininet.dll is in CAB16 on the ME CD. You can extract it using Explorer. Extract it to a \TEMP location or a floppy.