How do I mediate disputes where both sides are dug in?

You can settle most disputes that come to you, but the entrenched, high-conflict ones still defeat you. That is not a shortfall in your skill. Those cases resist everyone, because the people in them are often more committed to the fight than to any resolution. In the hardest disputes, agreement may not be the honest goal, at least not at first. What you can offer is a process that stays fair and calm even when the parties will not. Sometimes the win is not settlement but keeping the conflict from getting worse while something slowly shifts. With your next entrenched case, aim first to lower the temperature rather than reach a deal, and find the one small thing both sides can agree on. Build from there. You cannot force people out of a fight they are attached to, but you can be the steady ground under it.