As I understand your question, you wish to have a child together and start a family. This means, that both of you will be parents to your rainbow child. You have tried home-insemination without luck.
Like all other couples that wish to have a child, you both need a fertility investigation. This investigation does not differ from heterosexual couples. If you live in Denmark, you can ask your GP for a referral. After the investigation, we will be able to counsel you about the different treatment options (insemination, IVF or gamete donation).
Since you are 37 years old, I will recommend that you start with the fertility investigation now. You can always decide later on, whether you want to continue with home-insemination once you know the results, or if you want to start fertility treatment.
Best wishes, Ursula Bentin-Ley