I have such a serious patient in my family to take care of that I must keep an eye on her (him) at all times..
I have such a serious patient taken care of in my family that I'm tired from keeping an eye on her (him) at all times.
様々な表現が可能と考えますが、下記のように素直に表現してみました。
I have such a serious patient in my family to take care of that I must keep an eye on her (him) at all times.
私には家族の中に面倒を見なければならない重病患者がいるので、常に目を光らせていなければならない。
I have such a serious patient taken care of in my family that I'm tired from keeping an eye on her (him) at all times.
私には家族の中に面倒を見なければならない重病患者がいるので、常に目を光らせていなければならないので疲れています。