I know this is an old thread, but i believe it can still be helpful to post solutions here in case it helps someone like me who is searching for an answer.

I was having a similar issue:

Received the "No Emission Current" Error indicative of a blown filament. Vented the 5975C, replaced both filaments and pumped down.

Neither filament would light, but I was not getting errors from the unit. It would run Air & Water Checks / Autotune, but would not light the filament and therefore had no signal (8 counts on the A&W check for m/z=28).

Things I tried based on the comments here:
Reverted to an old atune.u in case my file was corrupted: No Change.
Replaced the foreline pump (had been planning on this anyway, switching to a scroll pump): No Change.
Calling Agilent: No help!

Somewhere, possibly here, there was a mention of a short circuit stopping the unit from applying current to the filaments w/o an error appearing.

I removed the newly installed filaments and found that one had a manufacturing defect where one pole was not secure in the ceramic insulator. This allowed it to move when the lead was connected and it was contacting the ion source body.

I replaced the filament and am back to 100% working order.