May HaShem grant you successful surgery and recovery. Presumably, after receiving instructions and gaining experience dressing and undressing in this situation, you will be able to safely do more than you now think. However, we will assume that, for at least some time, you will not be able to roll up the sleeve on your left arm and/or remove the sling.
Tefillin are supposed to be placed on one’s weaker arm.[1] There is an opinion that when the weaker arm is unavailable, one puts the tefillin on the stronger arm without a beracha.[2] However, the consensus is that the right arm is not an option for a right-handed person.[3] We will therefore investigate only the options for your left arm.
Is it absolutely necessary for the tefillin and/or the retzuot to be placed directly on the body? The mishna[4] criticizes one who puts tefillin shel yad on his sleeve. Rashi explains that such a person believes that since the Torah[5] calls tefillin an ot (sign), it should be visible to others. However, we expound the pasuk that “it is an ot for you – and not for others.”[6] The simplest understanding of Rashi[7] is that the tefillin shel yad should not be put on in a way that implies that it is primarily directed toward others. The Rosh[8] understands that, given that the tefillin shel yad do not require being exposed, they must be placed directly on the skin, thus precluding a chatzitza (separation) between the arm and the tefillin.
How broadly does the matter of chatzitza affect tefillin? The Rashba expresses the conviction that it does not apply to the shel rosh[9] or the retzuot, even even those of the shel yad.[10] However, he concludes[11] that the practice is to be careful so that the shel rosh, the shel yad, and the retzuot are free from any chatzitza. The Rama[12] rules that chatzitza is not a problem for the retzuot, but many Acharonim[13] take issue with this claim, at least in regard to the part of the retzuot that are needed to tie the tefillin down.
If the main problem with the tefillin being on top of something else is that it becomes an ot to others rather than to oneself, then it is possible to suggest some distinctions and points of leniency. First, when there is a physical need to have the arm covered, perhaps it is acceptable as long as the tefillin is covered with another layer.[14] There is also room to distinguish between a bandage or cast, which introduce a chatzitza, and a sleeve, which makes the tefillin considered as being on the outside of the arm. However, the Chayei Adam,[15] cited by the Mishna Berura,[16] rules that the tefillin cannot be on top of a sleeve even if there is another covering above the tefillin and even if the sleeve is medically required. While it is possible to dispute this logic and allow someone who cannot remove the sleeve to put the tefillin on top of it, one should not do so with a beracha on the tefillin shel yad.[17] Also, if one can manage to place the tefillin directly on the upper part of the bicep,[18] he can rely on the Gra’s view that that is also a valid spot for the tefillin shel yad.[19]
If it is possible to put the bayit of the tefillin and the part of the retzuot that fastens it on the arm itself, the fact that the remainder of the retzuot are on the sleeve is less of a problem. In your case, it seems that the forearm can be exposed and the question is whether you can get the sleeve off without hurting yourself,[20] which would be the best thing. The ability to get seven wrappings on the forearm is not a major issue and certainly does not preclude making a beracha, and thus the sling does not create a serious halachic problem.[21]
If you are unable to put on the shel yad, you should still put on the shel rosh. An Ashkenazi would make both berachot on the shel rosh, while a Sephardi would make only “…al mitzvat tefillin.”[22]