convert an illustrator file from rgb to cmyk for print

The file is RGB Photoshop but must be CMYK Photoshop for print. When we convert to CMYK the color is knocked flat and becomes very dull. We need this file in CMYK BUT it must be an *exact* color match. *Please can you include a step by step description of what you did to achieve this.* Three files are included: 1) the "compare RGB to CMYK" eg jpg, that shows the color loss when we convert to CMKY. We need the file to be CMKY BUT look exactly like the RGB globe on the left in this example. 2) The logo currently as an RGB jpg. Please convert this file to CMYK. 3) Also included is the original AI file, which you will see is in CMyK but also has the color as washed out and flat too. This file is included in case it is of assistance to you."