Logo Rebuild for Large Prints

I need my existing logo rebuilt from scratch so it can be used on banners and other large-format collateral without any loss of quality. Right now I only have a small, slightly blurry JPEG; no original vector file is available... (Budget: $10 - $30 AUD, Jobs: Adobe Creative Cloud, Adobe Illustrator, Adobe Photoshop, Graphic Design, Illustrator, Logo Design, Photoshop, Vector Design)