Low-res logo image redrawn as vector

Design details: I have a low res, one colour, image of logo file, the only remaining version available from my client.
I need a vector version as close as possible to the 'impression' of this low-res image.


I will need to open this in Illustrator, so an ai or an eps file return.
Ideas for the visual style: one colour illustration
Where the design will be used: print and web
Extra notes: I am attaching the file I referred to above.
I do not need the text as part of the finished item. I am hoping I can find the font.
I am anticipating that this might take a skilled illustrator about an hour.