I'm not sure if this will work with you but when my kernel boots it says what my soundcard is. It's a sounblaster 16 isa pnp card. If you type dmesg | more and read through it carefully you might see something that will indicate what card it is. If you know what your motherboard is you could look up the documentation. Also if you have windows on your machine (don't be offended that I mentioned that as a possibility
it will probably say what card it is. As a last resort i'm sure it's possible to get sndconfig working on slackware somehow.
As for the kernel, I've never used the kernel patches before so I'm not the person to talk to. As I understand it that is a little bit more compilcated since you have to patch the kernel source into the old source. I don't know which version you need to have of the old source though, it might have to be a specific one. Like I said though, I'm not familiar with that process so maybe someone else can help.