My laptop is Lenovo flex 2 pro 15 and I had the same problem.
I have modified these steps and for me works with live USB boot, persistence and version
# uname -a
Linux kali 4.17.0-kali1-amd64 #1 SMP Debian 4.17.8-1kali1 (2018-07-24) x86_64 GNU/Linux ----------> please make sure you have the same version
And you have the same hardware than me, using lspci
# lspci
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02:00.0 Network controller: Broadcom Limited BCM43142 802.11b/g/n (rev 01)
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Update and install necessary packages
# apt-get update
# apt-get install linux-image-$(uname -r|sed 's,[^-]-[^-]-,,') linux-headers-$(uname -r|sed 's,[^-]-[^-]-,,')
Download some extra packages, that do the trick for "wl module not found"
Code:
# wget http://http.kali.org/kali/pool/main/linux-kbuild-4.17_4.17.17-1kali1_amd64.deb
# wget http://http.kali.org/kali/pool/main/linux-headers-4.17.0-kali1-common_4.17.8-1kali1_all.deb
# wget http://http.kali.org/kali/pool/main/linux-headers-4.17.0-kali1-amd64_4.17.8-1kali1_amd64.deb
Install with correct order the .deb downloaded before,
# dpkg -i linux-kbuild-4.17_4.17.17-1kali1_amd64.deb
# dpkg -i linux-headers-4.17.0-kali1-common_4.17.8-1kali1_all.deb
# dpkg -i linux-headers-4.17.0-kali1-amd64_4.17.8-1kali1_amd64.deb
Install broadcom drivers
# apt-get install broadcom-sta-dkms
Enable modules and disabled unnecessary ones
# modprobe -r b44 b43 b43legacy ssb brcmsmac bcma
# modprobe wl
Now works fine