Fix broken Nethermind integration tests (#4836)

## Issue Addressed

CI is currently blocked by persistently failing integration tests.

## Proposed Changes

Use latest Nethermind release and apply the appropriate fixes as there have been breaking changes.
Also increase the timeout since I had some local timeouts.


Co-authored-by: Michael Sproul <michael@sigmaprime.io>
Co-authored-by: antondlr <anton@delaruelle.net>
Co-authored-by: Jimmy Chen <jchen.tc@gmail.com>

.github/workflows/test-suite.yml

@@ -317,10 +317,11 @@ jobs:
           ./doppelganger_protection.sh success genesis.json
   execution-engine-integration-ubuntu:
     name: execution-engine-integration-ubuntu
-    runs-on: ubuntu-latest
+    runs-on: ${{ github.repository == 'sigp/lighthouse' && fromJson('["self-hosted", "linux", "CI", "small"]') || 'ubuntu-latest'  }}
     steps:
     - uses: actions/checkout@v3
     - name: Get latest version of stable Rust
+      if: env.SELF_HOSTED_RUNNERS == 'false'
       uses: moonrepo/setup-rust@v1
       with:
           channel: stable
@@ -328,6 +329,9 @@ jobs:
           cache: false
       env:
           GITHUB_TOKEN: ${{ secrets.GITHUB_TOKEN }}
+    - name: Add go compiler to $PATH
+      if: env.SELF_HOSTED_RUNNERS == 'true'
+      run: echo "/usr/local/go/bin" >> $GITHUB_PATH
     - name: Run exec engine integration tests in release
       run: make test-exec-engine
   check-code: