I've had this plant for twenty years. I would love to know it's name! It is indestructible, which is a good thing because I am not a very attentive plant owner. It has in the past gone dormant for months, but it has always come around. Several times it has put forth a beautiful long shoot topped by a large ball of tiny orange flowers with stamens full of pollen that is very good at staining anything it touches. At the moment, as you can see, it's, or rather they, are sending up new shoots. I'm wondering if I should chop off the old growth now or later (I've always waited until the old leaves have begun to die back), but perhaps the new growth would benefit if I cut them back now. Any information would be greatly appreciated.
From your description of the flower, I think this is most likely a Scadoxus. I am not sure which species you have but Scadoxus multiflorus is probably a good candidate. There are a few subspecies, but I think Scadoxus multiflorus subsp. katharinae and Scadoxus multiflorus subsp. multiflorus are most common in cultivation.
If I was you I would not cut the plant back yet; it looks fairly healthy but a bit thirsty- both the growing medium and the drainage saucer look dry. I would probably just give it a deep watering. If you need to, take it outside or put it in the bath or shower so you can give it a really good soak and let it drain without flooding your floor!