As said - depart Southampton on a published SID. Have a look at the EGHI textual data in the online AIP:
L9/L10 GWC - SFD - R803 - DVR
You can then pick up the ALKIN3F from WAFFU into DET and then direct LAM. (They may let you go SFD-BONDY-DET if you are lucky)
HOWEVER - you may descended out of CAS if there are City/Biggin inbounds and outbounds to conflict as you get to DET.
As I said, you have to fly the SID to meet the STAR - nippy bizjets do it and can be hard work - just imagine it in a PA28.
Unfortunately for you, some inconsiderate berk put Gatwick/Heathrow/City/Biggin Hill in the way.