It's not worth saving this houseplant. Looks like a pothos and can easily and cheaply be replaced. While you're at it, replace all that old potting soil. Scrub out the pot. This old soil could be full of salt and disease.
I understand your take on this, however it has a lot of sentimental value. It was given to me at my husband‘s funeral and I do plan to save it however bad it looks. Just needed some advice to try to save it.
We understand. The best approach to saving it would be through propagation rather than repotting, in case too many old roots have been lost. You can certainly repot what's left of the rootball once cuttings have been taken, to see if it rebounds; instructions for this are below.
This plant roots easily at nodes - these are the joints in the stem where leaves are typically attached. Often, these same joints can be stimulated to also produce roots when kept moist. You can clip the stem into sections; length may not matter much, but if you want more chances at rooting success, use smaller pieces so you can get more of them. The nodes are where roots will emerge, not the internodes (section of stem between them), so the important part is to clip the stem in such a way that at least one or two nodes are included per piece. You can either keep the cut ends with one node under water (like in a vase or cup) or in soil (potted in a small plant pot with drain holes). Keep the water freshened if it starts to get murky or green, but otherwise, plain tap water is fine. They may take a few weeks to grow roots, but sometimes it can occur more quickly. Roots will look a bit like a slender stem, but will be creamy white. While the cutting is rooting, try to keep the leaves, if any (it's ok if the cutting has no leaves), out of direct sunlight and in a humid area, though pothos are fairly tough plants and are tolerant of lower household humidity levels. If it's easier to follow a tutorial, there are many videos on sites like YouTube on houseplant propagation. You can search for "propagating" or "cuttings" and "pothos" to get those more relevant to your situation. Once roots have grown, you can transfer the cutting to a pot and cover the root(s) with soil. Try not to use too large a pot in proportion to the size of the cutting so the soil doesn't stay soggy for too long. As far as soil type, regular potting soil is fine; it's probably best to avoid those that are labeled as having extra water-holding capacity or with included fertilizers. If you can't find any that fit this, a "seed-starting mix" should work as well.
Videos online will also give tutorials on repotting, but in essence: take off all the old soil and replace it with fresh potting soil. To remove old soil, it can be easier when the soil gets fairly dry first, as it will be easier to shake off in to a trash can. Pothos roots are not very fibrous, though, so old soil will probably fall away fairly easily. You can rinse off anything left in a sink or bucket. Then, take a fresh pot (or the cleaned same pot) and fill-in around the plant's roots with fresh soil. Set the cutting (or old root ball) so that the soil comes just up to the top of the roots, but doesn't cover-up the stems. Don't press it down too firmly, but you can tap the pot on the table to get the soil to settle-in around the roots so there are no major air gaps. Gently watering the pot will also get the soil to settle without compressing it too much. If you use a saucer to collect drips underneath the pot, just make sure it doesn't hold a puddle of water after the plant is drained or it can over-saturate the soil. Drip-drying a plant over a sink before putting it back onto a saucer is often the easiest way to water a plant without risking this.
Pothos thrive in moderate light (they tolerate low light, but in the long term this will starve the plant of energy) and soil that drys out fairly well between waterings. They do not like to stay constantly damp. As they grow and trail, if stems get too long, you can clip them back and propagate the clippings again or discard them. The newly-potted plants can be fertilized a few weeks after potting if you wish, though we do not recommend continuing to fertilize in winter, as indoor plants are fairly dormant during this time and do not need the supplemental nourishment. Fertilizer choice depends on which is easiest for you, but a general-purpose houseplant fertilizer (in other words, not bloom-boosting or for any other particular purpose) will be fine, whether in liquid or pellet form. Most of the time, less is more with regards to indoor plant fertilizing.