How to increase write speed with USB High speed host mode with STM32F4?
The library linked in your question does not support USB High Speed mode.

Rather, it supports use of the on-chip High Speed OTG peripheral in Full Speed mode.

Supported USB FS and USB HS in FS mode
  USB OTG is connected to USB FS mode on STM32F4-Discovery board
  USB OTG is connected to USB HS in FS mode on STM32F429-Discovery board


Some "reading of the fine print" is necessary to see this in the above - specifically that HS is mentioned only when followed by in FS mode.  And in fact the word "high" appears nowhere on the main page of your link.

It's also quite logical why this is the case: the hardware will not support high speed operation without the use of an external PHY chip connected to the processor's ULPI pins.  The wiring connections given in the documentation of the software library are specifically for connection to one of the on-chip full speed PHYs.

The STM32F4DISCOVERY (which actually has an 'F407 not an 'F405) does not contain a high speed PHY.  It breaks out the ULPI pins, but you would have to provide a carrier board containing the PHY chip.  (Also note that on this board the software is using the full speed peripheral engine - only on the `F429 does it use the high speed peripheral, but still in full speed mode).

Its likely that trying to add a PHY is not going to be a wise choice to achieving your ultimate goal, especially as there would be a lot of software work to be done.  You may want to try switching to another MCU (though high speed PHYs are more commonly found on SoCs than MCUs), and perhaps do some research on comparative performance that might be achieved with one able to operate a high performance SD card in native (rather than low performance SPI) mode.  In reality though, any path to achieving high speed logging other than buying a proven hardware + operating system solution is likely going to be a lengthy and challenging technical undertaking.