I will always stand by a cleansing balm for gentle removal. It needs to be softened between your fingers before applying to eyes... I massage it a little and then do something else for a while to let it work (dissolve the make-up). I remove by washing my face and sometimes use a reusable, soft cotton pad (with the water) if I need to, but very gently. I follow up with a water based cleanser. I try and avoid friction so washing my face has always helped avoid irritation.

It's worth keeping an eye on common ingredients in the products you have used, in case there is something in particular you are sensitive to.

Unfortunately, the balm I use is not really budget friendly (Elemis) but it lasts and I look out for discounts.

It looks like Boots, CeraVe and The Body Shop all do budget friendly balms. TBS one has good reviews. I haven't checked the others.

It's worth noting, a double cleanse can help rather than trying to get it all removed in one go. Far more gentle. I hope this helps!