It is possible to have joy in the midst of and as a result of deep sorrow. This comes from a deep relationship with the Lord and by the power of the Holy Spirit. It certainly does not come from our own efforts or inclinations.
I remember the deep sorrow of watching my first husband, Wayne, deteriorate in his health step by step. I remember the anxiety and concerns of how best to address his needs during that time. I remember the discouragement of knowing he would not be able to ultimately win the battle against his disease on this earth. Yet I also remember this as a time when I drew very close to the Lord. The intimacy of that relationship brought joy and comfort to me. The same was true when I lost Wayne to death. The Lord and the promises of His Word became increasingly precious to me. Also opportunities for minister to other people came about because of the sorrows I had experienced. This added much joy to my life.
As I have written before, I recently remarried after over eight and a half years of widowhood. I can tell you that it is true that the joys of the highs and the thankfulness for God's blessings are better because of the past sorrows. There are adjustments in a new marriage, but one looks at even these minor things in the light of God's love. There is also a family member who is currently going through some difficult struggles. This family member is someone for whom I have deep concerns and for whom I sometimes grieve. Yet in the Lord and His blessings, there is a deep joy that only He can give.
Whether you are a caregiver or have lost a dear loved one or whether you are going through some other deep heartache and sorrow, run to the Lord. He will give you a measure of His peace and joy even in the midst of the sorrow. He will also use that sorrow to bring you great joys in the future.