I think that your reasons for getting the birthday child's party day mixed up are very understandable, and I think it would still be understandable if, after the week you had had, you didn't feel up to shopping for a present and card for said child, and/or you didn't feel up to actually taking your child to the party at that particular time.
In your OP @Aintgotnomama
you told us that in your quick response to the b.day child's mother's comment on Facebook, you didn't explain everything to her in that message, as you thought she might be too busy to read it all at that time. I still think that you should have sent a full explanation, including your 2 day stay in hospital and why, and then the Mum could have decided how much she wanted to read at that time. I really do think that that is almost certainly the crux of the matter. The child's mother isn't a mind reader, so to her you just sent her a message - sometime after the party - saying something like "Oh no, I'm sorry Delilah, I have just seen your Facebook comment, and I have realised that I got the days for the party mixed up, I thought it was tomorrow! So please accept my apologies, and also tell Penelope that I am very sorry. Please also wish Penelope a slightly belated Happy Birthday from me, and from Sammy, who is upset that she missed your party".
Please excuse me now OP, as I am going to allot the main characters here some pen names, in order to hopefully help both you and me from getting too confused! I think if I had been in Delilah's (the bday child's Mum) place, that I would have replied with something along the lines of, "yes, that is a pity, as Penelope (the bday girl) was quite upset that Sammy didn't turn up. However, I will try to explain to her that you got your days mixed up". But I'm afraid that I can understand if Delilah couldn't even be bothered to give you a basic reply to that rather lackluster apology. As far as Delilah was/is concerned, you have never explained - your very good reasons - for getting the dates mixed up, and it would have been crass of her to ask you why. But, I think that what would have upset me the most if I was Delilah, was that you couldn't be bothered to give my dear child a card and a present, a £5 present, wrapped nicely would have been enough, and much appreciated by a young child.
Deciding to keep the explanation for your confusion to yourself, until the Mum replied to you, was in my opinion really not a good idea, and neither was not dropping the card and present off on the Sunday, when you thought the party was happening, or at the very least taking the present to the school with you on the Monday, to give to the child's Mother. So, again, in my opinion, that was rather rude of you, unless your child's birthday party was going to be on that same Monday, and even then, giving the child (Penelope) the present at your own child's party, would probably not seem as special to her, as receiving it when it was just about her.
I am only giving you my thoughts on why I think that the bday child's mother could be thoroughly fed up with you, because after quite a few pages, you still don't seem to be aware of what has annoyed the other Mum so much, and you appear to want to understand? So, I have shared why I would have been cool with you after your short, and uninformative message. However, I would not have growled at you, or even been rude to you, but I wouldn't have been able to not be cool with you, as I would - not having been given any explanation yet about why you got so mixed up - have still been disappointed on behalf of my child, and surprised by your lack of good manners.
Part of my reasoning above has been taking into account that as the party was at a soft play venue, I have guessed that the children involved were quite young, at least under the age of 8 years old. My apologies if you have already given the children's age, I must have missed that part.
Please @Aintgotnomama, send the explanation now , however belated it is, and give her child her present ASAP. I would also give the mother the money that she would have had to pay out for your child to go to the party, or, preferably in my opinion, I would give the mother a thoughtful little gift, in leu of paying for how much she was out of pocket because your child didn't attend the party. After doing all of that, the mother (Delilah) would hopefully start to thaw out, but if she didn't, you would have tried your best, in which case I would just say hello if our paths crossed, and then put her out of my mind.
I am sorry if you feel that I have been too harsh on you OP, but I do just want you to know how, or why, your problem with the bday child's Mum, might have happened. I truly hope that your life will improve very soon, and that you don't have any more flare ups of your nasty sounding condition, Take care, and good luck OP 💐